MD Jan Kulig
email: mskulig@cyf-kr.edu.pl

M Sierzega

P Kolodziejczyk

J Dadan

M Drews

M Fraczek

A Jeziorski

M Krawczyk

T Starzynska

G Wallner

Implications of overweight in gastric cancer: a multicenter study in a Western patient population

Keywords: gastric cancer, overweight, body mass index, lymph node dissection, prognosis, postoperative complications

come

INTRODUCTION

Obesity is gaining increasing attention in surgical oncology. This is partially due to the recent epidemiological data suggesting that excess body weight is associated with an increased risk of various gastrointestinal malignancies, including gastric cancer, and particularly oesophagogastric adenocarcinomas. [START_REF] Renehan | Body-mass index and incidence of cancer: a systematic review and meta-analysis of prospective observational studies[END_REF][START_REF] Yang | Overweight, obesity and gastric cancer risk: Results from a meta-analysis of cohort studies[END_REF] Moreover, from the surgical point of view, overweight patients are also exposed to a greater risk of interventions due to obesity-related comorbidities and increased incidence of postoperative complications. Another important aspect is related to some previous studies suggesting that overweight may influence the oncological outcomes of patients with gastrointestinal cancers. [START_REF] You | Effect of body mass index on the outcome of patients with rectal cancer receiving curative anterior resection: disparity between the upper and lower rectum[END_REF][START_REF] Moon | Visceral obesity may affect oncologic outcome in patients with colorectal cancer[END_REF] Several authors have already reported that overweight increases the incidence of general and intra-abdominal complications in Asian patients undergoing gastrectomy for gastric cancer. 5- 12 Some recent studies also suggested that obesity may correlate with long-term prognosis. [START_REF] Dhar | Body mass index determines the success of lymph node dissection and predicts the outcome of gastric carcinoma patients[END_REF][START_REF] Moriwaki | Does body mass index (BMI) influence morbidity and long-term survival in gastric cancer patients after gastrectomy?[END_REF][START_REF] Tokunaga | Better 5-year survival rate following curative gastrectomy in overweight patients[END_REF] However, as the prevalence of obesity is one of the principal differences between Western and Japanese or Korean patients with gastric cancer, influencing surgical technique and staging, simple approximation of observations reported from Asian populations is biased. [START_REF] Bunt | Lymph node retrieval in a randomized trial on western-type versus Japanese-type surgery in gastric cancer[END_REF][START_REF] Davis | The difference in gastric cancer between Japan, USA and Europe: what are the facts? what are the suggestions?[END_REF] This is particularly important for lymphadenectomy, as excess abdominal fat may markedly impair lymph node dissection and can play an important role in explaining the differences in surgical outcomes between Western and Asian patients. Surprisingly, data for the association between overweight and long-term outcomes outside of Japan and Korea is scarce. Moreover, the paucity of studies concerning Western populations and potential flaws in reports from single institutions or including only a limited number of patients preclude a reliable evaluation of existing data. [START_REF] Barry | Western body mass indices need not compromise outcomes after modified D2 gastrectomy for carcinoma[END_REF] The aim of this study was to examine the relationship between overweight and long-term survival of patients undergoing gastrectomy for gastric cancer and to evaluate the potential mechanisms underlying these effects in terms of postoperative complications and adequacy of surgical therapy.

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METHODS

Patient selection

An electronic database of all patients with resectable gastric cancer treated between 1986 and 1998 at seven university surgical centres participating in the Polish Gastric Cancer Study Group (PGCSG) was reviewed. All relevant data, including demographics, clinical findings, details of surgical procedures and histopathological parameters, were collected prospectively using standard forms. Overweight was defined according to the World Health Organization classification as a body mass index (BMI) of 25 kg/m 2 or higher. [START_REF] Seidell | Assessing obesity: classification and epidemiology[END_REF] This study was approved by the Bioethics Committees at each institution.

Treatment and follow-up

The type of stomach resection was selected based on tumour location and was not influenced by patients' BMI. In general, a subtotal gastrectomy with Billroth II reconstruction was used for tumours of the distal and some middle stomach. Proximal gastric cancer without gross involvement of surrounding tissues was treated with a proximal subtotal gastrectomy and oesophagogastric anastomosis. Total gastrectomy with Roux-en-Y reconstruction was typically used in other cases. Primary tumours were resected en bloc with perigastric lymph nodes (groups 1-6 according to the Japanese Gastric Cancer Association (JGCA)), whereas extragastric nodal stations (groups 7-11 and/or 7-12, depending on the site of the primary tumour) were dissected separately. The extent of lymphadenectomy was described by the operating surgeon and finally evaluated by the pathologist by counting resected lymph nodes in individual stations as defined by the JGCA. Lymph nodes were identified and retrieved from surgical specimens without using any specific technique to increase nodal retrieval rate.

Paraffin-embedded nodes were stained with haematoxylin and eosin, and examined microscopically for metastases. No special immunohistochemical techniques were used to

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identify micrometastases. Follow-up data were collected based on clinical examination every 3-6 months after discharge and dates of death were verified using data obtained from the census registry office. Survival rates were calculated based on the disease-specific principle, i.e. deaths due to gastric cancer were accepted as complete observations, while death due to other reasons and lost to follow-up was considered as a censored observation.

Statistical methods

Mann-Whitney U and χ 2 tests were used where appropriate to compare the distribution of individual variables between groups. Survival data was analysed according to the Kaplan-Meier method and included postoperative mortality; the log rank test was used to detect differences between groups. Multivariable analysis was performed using a Cox proportional hazards model with a backward stepwise selection procedure. The probability for entering the model was 0.050 and that for removal from the model was 0.100. P < 0.050 was considered statistically significant. Statistical analysis was performed using the SPSS ® version 16 software package (SPSS, Chicago, Illinois, USA).

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RESULTS

Demographic data

Between 1986 and 1998, 1992 records with detailed information regarding patient-related and tumour-related variables, including body height and body weight, were identified in the PGCSG database. The median BMI was 22.7 (range 15 to 39) and 492 patients had BMI of 25 kg/m 2 or more. Table 1 shows clinico-pathological features of the analysed population.

Compared to patients with BMI <25 kg/m 2 , overweight was associated with a slightly higher incidence of cardiocirculatory disorders and a significantly higher rate of diabetes mellitus.

Surgery

No significant differences were observed with respect to the type of gastrectomy, extent of lymphadenectomy, and the proportion of patients undergoing pancreaticosplenectomy.

There was a negative linear correlation between BMI and the number of dissected lymph nodes with a correlation coefficient r = -0.072 (P=0.002). Overweight patients had a higher proportion of cases with 15 or less nodes in the surgical specimen (45% vs 37%, P=0.002) and the correlation between this proportion and BMI was highly significant (r=0.759, P<0.001). The overall mean number of removed lymph nodes in our series was lower than currently accepted as a standard, but most patients analysed in this study were operated on before 1995, when a policy of routine extent of lymphadenectomy of at least D2 was implemented. Moreover, a limited lymph node dissection was generally performed in all patients with synchronous distant metastases. The following variables were tested by logistic regression analysis to identify those associated with an inadequate number (i.e. D1 lymphadenectomy (Odds Ratio (OR) 4.16, 95%CI 3.12 to 5.54, P<0.001), absence of lymph node metastases (OR 2.13, 95%CI 1.64 to 2.75, P<0.001), tumours limited to the gastric mucosa and submucosa (OR 1.57, 95%CI 1.05 to 2.35, P=0.027), and BMI (OR 1.03, 95%CI 1.01 to 1.07, P=0.033) were independent predictors of low lymph node count in surgical specimens.

Pathology

The anatomical distribution of tumours within the stomach and the proportion of individual Lauren's types were similar between groups. Overweight patients had a significantly lower proportion of category T3 and T4 tumours (64% vs 71%, P=0.005) and the prevalence of synchronous distant metastases. Moreover, patients with a BMI below 25 kg/m 2 were significantly more likely to undergo non-curative resection.

Overweight was associated with a lower proportion of patients with metastatic lymph nodes (59% vs 70%, P<0.001) and a lower median ratio of metastatic/resected lymph nodes (0.17 vs 0.30, P=0.003). For both of these parameters there was a significant negative linear correlation with BMI and corresponding Pearson's coefficients of regression were -0.730 (P=0.003) and -0.075 (P=0.007), respectively. However, if 16 or more lymph nodes were dissected, the proportion of patients with individual N categories was unaffected by BMI and there was no correlation between BMI and the prevalence of metastatic lymph nodes or the ratio of metastatic/resected lymph nodes.

Postoperative course

Postoperative recovery was uneventful in 1277 patients and morbidity rates for overall complications were similar regardless of BMI (Tab. 2). Overweight was associated with

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significantly higher rates of cardiopulmonary complications (16% vs 12%, P=0.010) and intra-abdominal abscess (6.9% vs 2.9%, P<0.001). Other complications and mortality rates were not influenced by BMI. The median postoperative stay was longer in overweight subjects (17 vs 14 days, P<0.001).

Long-term survival

At the time of final follow-up (December 2006), 1295 of 1992 patients had died, and the median follow-up period for surviving subjects was 104 months. The median disease-specific survival of overweight patients was significantly higher (36.7 months, 95%CI 29.0 to 44.4) than those with BMI<25 kg/m 2 (25.7 months, 95%CI 23.2 to 28.1; P=0.003) (Fig. 1). To explain these differences, survival rates between overweight and non-overweight patients were compared after stratification based on tumour staging. Since Kaplan-Meier's curves for groups with BMI <25 kg/m 2 and ≥25 kg/m 2 superimposed on stages I-II and III-IV (Fig. 2A-B), survival benefits of overweight were related to the differences in tumour staging.

A Cox proportional hazards model was constructed including variables that significantly affected disease-specific survival in the univariate analysis, i.e. age (<40, 40-65, >65 years), BMI (<25 kg/m 2 , ≥25 kg/m 2 ), tumour location (upper, middle or distal third, whole stomach), Laurén's classification (intestinal, diffuse, mixed), depth of infiltration (T1-T4), nodal status (N0-Nx), metastatic/resected lymph node ratio (0, 0-20%, >20%), distant metastases (no, yes), potentially curative resection (no, yes), splenectomy (no, yes), and postoperative chemotherapy (no, yes). The only independent prognostic factors were age, depth of infiltration, lymph node metastases, metastatic/resected lymph node ratio, distant metastases, and residual tumour category (Tab. 3). The hazard ratio in overweight patients compared to those with BMI <25 kg/m 2 was 0.89 (95% CI 0.77 to 1.04, P=0.139). Considering the differences observed among tumour stages, two additional Cox models were constructed for

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stages IA through II and separately IIIA through IV. Nevertheless, overweight was not a significant predictor of survival with hazard ratios of 0.940 (95%CI 0.703 to 1.257, P=0.667) and 0.916 (95%CI 0.777 to 1.079, P=0.292), respectively. Similar results were obtained if the model was stratified for curative and non-curative resections.

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DISCUSSION

The influence of overweight on survival

One of the major clinical differences between patients with gastric cancer in the West and those in Japan or Korea is a much higher prevalence of overweight and obesity that may have important practical consequences. The most important aspect is related to findings from Japanese populations suggesting that obesity may correlate with long-term prognosis of gastric carcinoma. Tokunaga et al. [START_REF] Tokunaga | Better 5-year survival rate following curative gastrectomy in overweight patients[END_REF] presented the largest analysis to date evaluating the correlation of BMI and long-term survival in 7925 patients treated with curative intent in a Japanese centre. Unexpectedly, the 5-year survival rate of 84.0% in overweight patients (BMI ≥25 kg/m 2 ; n = 1126) was significantly higher (P<0.001) than 76.7% in the non-obese group (BMI <25 kg/m 2 ; n = 6799). Moreover, multivariate analysis of potential prognostic factors demonstrated that higher BMI significantly reduced the risk of death (hazard ratio 0.874, 95%CI 0.773-0.988, P=0.032). Despite these results, prognostic implications of overweight are not well established for gastric cancer as smaller Japanese studies demonstrated that high BMI was an independent predictor of recurrence [START_REF] Dhar | Body mass index determines the success of lymph node dissection and predicts the outcome of gastric carcinoma patients[END_REF] or questioned its influence on the long-term outcomes. [START_REF] Ojima | Influence of overweight on patients with gastric cancer after undergoing curative gastrectomy: an analysis of 689 consecutive cases managed by a single center[END_REF][START_REF] Inagawa | Effect of fat volume on postoperative complications and survival rate after D2 dissection for gastric cancer[END_REF][START_REF] Oh | Effect of being overweight on postoperative morbidity and long-term surgical outcomes in proximal gastric carcinoma[END_REF][START_REF] Kodera | Obesity and outcome of distal gastrectomy with D2 lymphadenectomy for carcinoma[END_REF] Similarly, the definitive implications of overweight in Western patients with gastric cancer have yet to be determined due to the paucity of data for the association between BMI and prognosis. [START_REF] Trivers | Demographic and lifestyle predictors of survival in patients with esophageal or gastric cancers[END_REF] Using a well defined multi-institutional data set from a Western country, the present study demonstrated that overweight was associated with a significant survival benefit by univariate analysis. However, unlike Tokunaga et al., [START_REF] Tokunaga | Better 5-year survival rate following curative gastrectomy in overweight patients[END_REF]

Correlation of BMI with postoperative complications

Most previous retrospective studies from Asian populations suggest that BMI influences the occurrence of surgical complications in gastric cancer surgery indirectly by increasing operation time [START_REF] Ojima | Influence of overweight on patients with gastric cancer after undergoing curative gastrectomy: an analysis of 689 consecutive cases managed by a single center[END_REF][START_REF] Dhar | Body mass index determines the success of lymph node dissection and predicts the outcome of gastric carcinoma patients[END_REF][START_REF] Inagawa | Effect of fat volume on postoperative complications and survival rate after D2 dissection for gastric cancer[END_REF][START_REF] Oh | Effect of being overweight on postoperative morbidity and long-term surgical outcomes in proximal gastric carcinoma[END_REF] and blood loss [START_REF] Ojima | Influence of overweight on patients with gastric cancer after undergoing curative gastrectomy: an analysis of 689 consecutive cases managed by a single center[END_REF][START_REF] Dhar | Body mass index determines the success of lymph node dissection and predicts the outcome of gastric carcinoma patients[END_REF][START_REF] Inagawa | Effect of fat volume on postoperative complications and survival rate after D2 dissection for gastric cancer[END_REF][START_REF] Hiki | Increased fat content and body shape have little effect on the accuracy of lymph node retrieval and blood loss in laparoscopic distal gastrectomy for gastric cancer[END_REF] , but also directly increasing rates of pancreatic fistula [START_REF] Kunisaki | Predictive factors for pancreatic fistula after pancreaticosplenectomy for advanced gastric cancer in the upper third of the stomach[END_REF][START_REF] Tanaka | Accumulation of excess visceral fat is a risk factor for pancreatic fistula formation after total gastrectomy[END_REF][START_REF] Ojima | Influence of overweight on patients with gastric cancer after undergoing curative gastrectomy: an analysis of 689 consecutive cases managed by a single center[END_REF] and abdominal abscess. [START_REF] Ojima | Influence of overweight on patients with gastric cancer after undergoing curative gastrectomy: an analysis of 689 consecutive cases managed by a single center[END_REF][START_REF] Katai | Risk factors for pancreas-related abscess after total gastrectomy[END_REF] Moreover, these observations are supported by the results of randomized clinical trials confirming the detrimental effects of overweight. [START_REF] Tsujinaka | Influence of overweight on surgical complications for gastric cancer: results from a randomized control trial comparing D2 and extended para-aortic D3 lymphadenectomy (JCOG9501)[END_REF][START_REF] Kodera | Identification of risk factors for the development of complications following extended and superextended lymphadenectomies for gastric cancer[END_REF] Although some reports from Western populations revealed that BMI was not a significant predictor of postoperative morbidity 18,[START_REF] Gretschel | Body mass index does not affect systematic D2 lymph node dissection and postoperative morbidity in gastric cancer patients[END_REF] , the present findings strongly suggest that overweight may significantly contribute to the early postoperative outcomes. However, as in other reports, high BMI did not increase mortality rates, and thus cannot be responsible for the marked differences in long-term survival rates.

Implications of overweight for lymph node dissection

Inadequate lymph node dissection during gastrectomy could be another factor potentially worsening a patients' prognosis, as excess fat may limit accessibility to lymph nodes located deep in the adipose tissue around major abdominal vessels not included in the routine en block resection. Such assumptions are based on several observations concerning the different extent of lymphadenectomy. [START_REF] Tsujinaka | Influence of overweight on surgical complications for gastric cancer: results from a randomized control trial comparing D2 and extended para-aortic D3 lymphadenectomy (JCOG9501)[END_REF][START_REF] Dhar | Body mass index determines the success of lymph node dissection and predicts the outcome of gastric carcinoma patients[END_REF][START_REF] Kodera | Obesity and outcome of distal gastrectomy with D2 lymphadenectomy for carcinoma[END_REF][START_REF] Hiki | Increased fat content and body shape have little effect on the accuracy of lymph node retrieval and blood loss in laparoscopic distal gastrectomy for gastric cancer[END_REF] However, previous reports from Western population-based databases and randomized clinical trials provided no data on the influence of BMI on the likelihood of undergoing adequate lymph node assessment. [START_REF] Bunt | Lymph node retrieval in a randomized trial on western-type versus Japanese-type surgery in gastric cancer[END_REF][START_REF] Coburn | Significant regional variation in adequacy of lymph node assessment and survival in gastric cancer[END_REF][START_REF] Baxter | Inadequacy of lymph node staging in gastric cancer patients: a population-based study[END_REF][28][29] In the present study, BMI significantly affected the proportion of patients with 15 or less lymph nodes removed and after stratifying patients for the extent of lymphadenectomy, this effect was preserved for both

Impact of BMI on tumour staging

Staging differences related to BMI may be another potential factor responsible for discrepancies in long-term survival between overweight and non-overweight patients. Present findings suggest that high BMI is associated with markedly reduced proportions of tumours infiltrating gastric serosa, surrounding organs, metastatic lymph nodes, and distant metastases.

As these variables are most commonly reported as independent prognostic factors for gastric cancer, the most likely explanation of better long-term survival found in the univariate analysis for overweight in our population is the correlation between BMI and tumour staging. This hypothesis is confirmed by the Kaplan-Meier's plots and Cox's proportional hazards models stratified for the American Joint Committee on Cancer (AJCC) stage (Fig. 2A-B). Data collected in a Japanese cohort of 7925 patients suggested a similar conclusion that the prevalence of T3-4 and N2-3 cancers decreases significantly in patients with BMI ≥25 kg/m 2 . [START_REF] Tokunaga | Better 5-year survival rate following curative gastrectomy in overweight patients[END_REF] Nevertheless, these correlations should be further validated with population-based databases since some studies involving smaller Asian populations failed to confirm any association between BMI and staging. [START_REF] Ojima | Influence of overweight on patients with gastric cancer after undergoing curative gastrectomy: an analysis of 689 consecutive cases managed by a single center[END_REF][START_REF] Inagawa | Effect of fat volume on postoperative complications and survival rate after D2 dissection for gastric cancer[END_REF][START_REF] Oh | Effect of being overweight on postoperative morbidity and long-term surgical outcomes in proximal gastric carcinoma[END_REF] Moreover, the mechanism of such less aggressive behaviour of gastric cancer in overweight individuals is yet to be determined, as some previous studies in hormone-dependent cancers (breast and prostate) and colorectal cancer reported a positive correlation between BMI and cancer stages, unlike the results of this study. 30

Conclusion

The present study provided for the first time data for the association between BMI and prognosis of gastric cancer from a large-scale Western-type population. Although higher BMI increased rates of some postoperative complications and impaired adequacy of lymph node dissection, better survival rates observed in overweight patients were related to the higher proportion of less advanced tumours. Therefore, overweight was not an independent prognostic factor for long-term survival.

  15 or less) of dissected lymph nodes: patient age (< 40, 40-65, > 65 years), sex, BMI (continuous variable), tumour location (upper, middle, distal third, whole stomach), Lauren's type, depth of wall penetration (T category), nodal status (positive or negative), distant metastases (present, M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 8 absent), type of surgical resection, pancreaticosplenectomy, and extent of lymphadenectomy.

  BMI ≥25 kg/m 2 was not an independent prognostic factor, suggesting that other covariates are responsible for such findings. The three most relevant aspects of overweight that may influence patients' survival are the relationship M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 12 between BMI and postoperative morality, tumour staging, and adequacy of surgical therapy, discussed below.

13 D1

 13 and D2 dissection. As several reports emphasised the importance of the stage migration phenomenon and poorer prognosis related to inadequate lymph node dissection, including our series (hazard ratio for Nx 1.85, 95%CI 1.48 to 2.32, P<0.001), this observation has important clinical implications for surgeons operating obese individuals.

21 TABLESTable 1 .

 211 AM, Haas O, Piard F, et al. How many nodes must be examined to accurately stage gastric carcinomas? Results from a population based study. Cancer 2002;94(11): 2862-2866. 29. Yang SH, Zhang YC, Yang KH, et al. An evidence-based medicine review of lymphadenectomy extent for gastric cancer. Am J Surg 2009;197(2): 246-251. 30. Calle EE, Thun MJ. Obesity and cancer. Oncogene 2004;23(38): 6365-6378. Clinico-pathological features of overweight (BMI ≥25kg/m 2 ) and non-overweight (BMI <25kg/m 2

Fig. 1 .

 1 Fig. 1. Disease-specific survival for overweight (BMI ≥25kg/m 2 ) and non-overweight (BMI <25kg/m 2 ) patients (p = 0.003, log-rank test).

Fig. 2 .

 2 Fig. 2. Disease-specific survival for overweight (BMI ≥25kg/m 2 ) and non-overweight (BMI <25kg/m 2 ) patients according to tumour stage. A. Stage I and II (p = 0.960, log-rank test); B.

Table 2 .

 2 Postoperative complications according to BMI

	005 †

Table 3 .

 3 Multivariate analysis of independent prognostic factors for survival (Cox

	proportional hazards model)
	Parameter	Category Hazard Ratio (95%CI)	P
	Age (years)	< 40	1
		40-65	1.35 (1.04-1.76)	0.024
		> 65	1.69 (1.26-2.27)	< 0.001
	BMI	<25 kg/m 2	1
		≥25 kg/m 2	0.89 (0.77-1.04)	0.139
	Tumour location	Distal	1
		Middle	0.99 (0.87-1.21)	0.881
		Upper	1.13 (0.66-1.81)	0.894
		Whole	1.16 (0.62-2.15)	0.647
	Laurén's classification	Intestinal	1
		Diffuse	1.01 (0.87-1.16)	0.932
		Mixed	1.02 (0.84-1.25)	0.809
	Splenectomy	No	1
		Yes	1.09 (0.95-1.26)	0.232
	Depth of infiltration	T1	1
		T2	1.94 (1.35-2.78)	< 0.001
		T3	2.71 (1.92-3.83)	< 0.001
		T4	3.33 (2.28-4.88)	< 0.001
	Lymph node metastases	N0	1
		N1	1.32 (0.96-1.81)	0.085
		N2	2.59 (2.10-3.18)	< 0.001
		N3	3.54 (2.75-4.56)	< 0.001
		Nx	1.85 (1.48-2.32)	< 0.001
	Metastatic/resected lymph node ratio	0	1
		0-20%	1.45 (1.07-1.97)	0.018
		>20%	2.11 (1.67-2.65)	<0.001
	Distant metastases	No	1
		Yes	1.37 (1.13-1.66)	0.002
	Residual tumour category	R0
		R1 or R2

ETHICS COMMITTEE APPROVAL

This study was approved by the Bioethics Committees at each institution.

ROLE OF FUNDING SOURCE

This study was financially supported by the Polish State Committee for Scientific Research, Grants no. 3 P05C 067 23p01, 2 P05C 024 29, 3 P05C 031 22, N403 069 31/3255

The funding source did not participate in study design, collection, analysis, or interpretation of the data, or writing of the report.

ACKNOWLEDGMENT

Other Members of the Polish Gastric Cancer Study Group:

AUTHORS' CONTRIBUTIONS

Study concept and design: Kulig, Sierzega, Kolodziejczyk.

Acquisition of data: Dadan, Drews, Frączek, Jeziorski, Kolodziejczyk, Krawczyk, Sierzega, Starzynska, Wallner.

Analysis and interpretation of data: Sierzega, Kolodziejczyk.

Drafting of the manuscript: Sierzega, Kolodziejczyk.

Critical revision of the manuscript: Dadan, Drews, Frączek, Jeziorski, Krawczyk, Kulig, Starzynska, Wallner.

All authors had access to portions of the raw data corresponding to their investigational sites. Jan Kulig, Marek Sierzega, and Piotr Kolodziejczyk had access to all data in the study and analysis.

CONFLICT OF INTEREST STATEMENTS

The authors declare that they have no conflicts of interest.


E M Bennani

Staicu Casagrande

Dal Cappello

C Li

A Lahmam-Bennani

C Dal

Electron impact double ionisation of neon, argon and molecular nitrogen: role of the two-step mechanism

HAL is

Introduction

In two recently published papers [1,2] Lahmam-Bennani and coworkers reported fourfold differential cross sections (4DCS) measurements for electron impact double ionisation (DI) of helium atom at an impact energy in the range E 0 ~600 -700 eV and for a large sampling of ejected electron energies. The main experimental finding in these works was that the experimental angular distributions of the 4DCS displayed large angular shifts of the forward and backward lobes with respect to the momentum transfer direction or it's opposite, respectively, as well as a breaking of symmetry with respect to these directions. A kinematical interpretation was given in [1] which allows relating the observed shifts and the existence of structures in the intensity distributions mostly to the second order, 'two-step 2' (TS2) DI mechanism, which was shown to predominate over the first-order 'shake-off' (SO) and 'twostep 1' (TS1) mechanisms under the considered kinematics. (For a discussion of the various DI mechanisms, see for instance [3][4][5][6][7]). This kinematical interpretation was nicely confirmed in [2] by the results of two model calculations which include non-first order mechanisms. The first model named TS2-MCEG is based on the first Born approximation (FBA), but where higher order effects are incorporated using the Monte Carlo Event Generator (MCEG) technique [8] to simulate the TS2 mechanism by convoluting two single ionisation (SI) events which are both calculated in the FBA. These are successively the SI of the He atom by the incident electron and that of the resulting He + ion by the intermediate scattered electron. The second theoretical model is more elaborate as it makes use of the second Born (B2) approximation and the closure approximation in the way described in [9]. Both these models were found to very clearly constitute a considerable improvement with respect to a first Born prediction as they succeeded in reproducing both the positions and the main structures of the forward and backward lobes.

The aim of the present work is to extend the work in [1,2] to other targets than helium (we use here two atomic targets, neon and argon, and one small molecule, molecular nitrogen N 2 ), in order to validate or invalidate the observations and the kinematical analysis therein given. Indeed, this analysis is merely dependent on the target species used. For given scattered and ejected electron energies, the target dependence does only enter via the small change in incident energy (< ~4%) due to the differences in the targets SI and DI potentials.

Our objective is to add further experimental evidence that at the impact energy of the present work (~600-700 eV), DI of small atoms or molecules is dominated by non-first order mechanisms such as TS2, similar to the Helium case. This gives a kind of "universal", target independent character to the observed shifts and positions of the 4DCS forward and backward lobes.

The present experimental data for Ne and Ar will be compared to first order theoretical results as well as to non-first order, second Born calculations where available. Due to the complex nature of the molecular target N 2 , no elaborate calculations are available at the time of writing.

Experiment

The experimental set-up and procedure used in the present work are identical to those used in [1,2] and are described in detail in [10]. Briefly, a well-collimated electron beam with energy E 0 ~600 -700 eV (see below) ionises a gas jet formed at the collision centre by the effusive flow of gas through a small capillary. Three high-efficiency, multi angle toroidal electrostatic energy analysers coupled to three position sensitive detectors allow observation of the final state electrons in the collision plane defined by the incident and scattered momentum vectors, k 0 and k a , respectively. The fast, forward-scattered electron (indexed 'a') is detected at the scattered energy E a = 500 eV and at two symmetrical scattering angles, θ a = + (6°± 3°) and -(6°± 3°) as set by input slits at the entrance to the 'a'-toroidal analyser.

Throughout this work, positive angles are counted clockwise starting from the incident electron beam direction. Among the two ejected electrons resulting from DI of the target, (labelled 'b' for the faster and 'c' for the slower), we choose to only detect the faster one, with energy E b , in coincidence with the 'a'-scattered electron, hence a so-called (e,3-1e) experiment. Of course, such distinction does not hold for the case of equal ejection energies, but the same labelling 'b' is kept for the detected electron. These 'b'-electrons are multi-angle analysed in a double toroidal analyser over the angular ranges θ b = 20°-160°and 200°-340°, where 0°is defined by the incident beam direction. In the off-line analysis, the total θ bangular range is divided into sectors of width ∆θ b = 5°. Though the emission direction of the third, 'c'-electron is unknown, its kinetic energy E c is known from the energy conservation E 0 -IP 2+ = E a +E b +E c , where IP 2+ is the target first DI potential leaving the ion in its ground state. (Here, it is assumed, following van der Wiel and Wiebes [11] that the DI process removing two electrons from the outermost orbital with the ion left in its ground state largely dominates DI processes with excited ion states). For each of the three considered targets (Ne, Ar and N 2 ), the experiments were performed at three ejected electron energies for the 'b'electron, E b = 12, 72 and 144 eV and a fixed 'c'-electron energy, E c = 12 eV, hence corresponding either to an equal or an unequal energy sharing (E b :E c ) among electrons 'b' and 'c'. We also note the very large range of energy loss (E 0 -E a ) suffered by the projectile, from 67 to ~219 eV. The incident energy (E 0 ) is consequently adjusted to fulfil the energy conservation requirement for the target under study, with IP 2+ = 62.6, 43.2 and 43.0 eV for Ne, Ar and N 2 , respectively. The investigated kinematical conditions are listed in Table 1. Given the above experimental parameters, the momentum transfer to the target, defined by K = k 0 -k a , varies in the magnitude from K = 0.76 au at E 0 = 567 eV to K = 1.39 au at E 0 = 718.6 eV, while its direction θ K (shown in the 4 th column of Table 1) varies from ~56°to 27°for these two extreme cases. Simultaneously, due to the ±3°acceptance in θ a -angle, the momentum transfer resolution amounts to ∆Κ ~± 0.2 au and the spread in the momentum transfer direction is ∆ K ~±10°.

E a = 500 eV θ a = -6 deg E c (undetected) = 12 eV Target E 0 (eV) E b (eV) detected θ K / θ -K (deg) θ F-TS2 / θ Β-TS2 (

Results and discussion

The angular distributions of the (e,3-1e) four-fold differential cross sections (4DCS),

d 4 σ /dE a dE b d a d b ,
for DI of the outermost orbitals of Ne (2p -2 ), Ar (3p -2 ) and N 2 (3σ g ) -2 are shown in figures 1 to 3, respectively at the three energy sharings (E b :E c ) listed in Table 1. As a general observation, we find for all targets and all energy cases about the same behavior as for He, i.e. all angular distributions display a forward and a backward lobe. Here, the labels forward lobe and backward lobe designate the lobe pointing roughly in the momentum transfer direction (+K) and in the opposite direction (-K), respectively. These two directions are indicated by the vertical dotted lines in the figures. We also observe that both lobes are not symmetrically distributed about ±K axis and that their maxima are shifted from this axis direction by large amounts, ~30°to 60°. The uncertainty of ~10°in the momentum transfer direction is significantly smaller than the observed shift, and hence it only marginally affects this shift. Also, these lobes exhibit more or less pronounced structures.

To better understand the origin of these features, the experimental data are compared with the calculated results obtained for Ne using the first Born -three Coulomb waves (B1-3C)

model. In this model, the initial state is described by a Hartree-Fock wave function of Clementi and Roetti [12]. The final state is described by the product of three Coulomb waves [13,14]. Two of these Coulomb functions describe each electron in the field of the target A strong disagreement is found between first Born calculations and experiments for Ne and Ar, the disagreement being appreciably more pronounced than was observed previously for He [1,2]. Indeed, for both atomic targets the B1-3C and B1-2CG models yield forward and backward angular distributions which are symmetrically distributed about ±K directions, as it is expected from any first-order theory, whereas the experiments do not show such symmetry.

Also, for the equal energy sharing case (Fig. 1(a) and 2(a)) the theoretical results display a marked minimum in the ±K directions which is not observed in the experiments. Moreover, the theory predicts a rather small backward intensity (relative to the forward one), smaller than the experimental findings. These observed deviations between experimental data and first order theory are clear evidence that strong non-first Born effects are present in the (e,3-1e) 4DCS distributions, that is, the contribution of the TS2 or higher order mechanisms to the DI process is sufficiently important with respect to that of SO and/or TS1 to impose its finger print on the angular distributions. We thus so confirm the observations made for He in [1,2] and earlier in [16]. However, the qualitative success of the second Born B2-2CG model reported for He is far from being found again here in the Ne and Ar cases. Indeed, when compared to B1-2CG predictions, the B2-2CG model does yield a breaking of symmetry with respect to ±K directions (at least in the case of Fig. 1(a) and 2(a)) as seen in the experiments.

But, if one excludes the difference in magnitude of the cross section (not relevant here), the small change in the shape of the angular distributions is not sufficient to bring the B2-2CG results in significantly better agreement with experiments than B1-2CG. The disagreement between our B2-2CG model and experiments here is certainly due to the approximations made for treating double ionisation of a noble gas. In this case we reduce the problem of a 6electron target (np 6 ) to that of two active electrons (the two ejected electrons) [17]. Moreover, Dal Cappello et al [9] have shown that the closure approximation used in B2 model yields results which strongly depend on the value of the parameter w used, where w is the average excitation energy. We notice that we used here the same parameter value for the cases of the double ionisation of noble gases as for helium.

We now aim to apply to our new results the kinematical analysis proposed in [1] and which has proven highly successful for the He case. This analysis is meant to give a qualitative interpretation of the observed structures as well as an estimate of the angular positions of the lobes in terms of physical effects included in the TS2 mechanism. The full details of this analysis have been previously reported in [1] and so are only briefly summarized here. TS2 is a two-step process, which is assumed to proceed via two successive (e,2e) ionisations of the target: in the first step, the 'c'-electron is ejected while the projectile is scattered with the highest probability at the ±θ a* angle corresponding to the Bethe ridge [18,19]. This intermediate 'a*'-scattered electron acts as a new projectile in a second (e,2e) ionisation of the intermediate, singly charged ion, resulting in the pair ('a':'b') of electrons effectively detected. These are the 'a'-scattered one, observed with E a = 500 eV under θ a = -6°, and the 'b'-ejected one which appears mostly along the corresponding momentum transfer direction of this new collision, either in the forward direction or in the backward direction, noted θ F-TS2 and θ B-TS2 , respectively, depending whether the intermediate 'a*'-electron is scattered under +θ a* or -θ a* .

The so predicted θ F-TS2 and θ B-TS2 values are listed in the last column of Table 1 for each of the kinematics considered here. They are also indicated in Fig. 1-3 by the vertical dashed lines. We have already pin-pointed the fact that all the experimental data shown in Fig. 1-3 do display prominent structures for both the forward and the backward distributions, though may be less so for the molecular target, N 2 . Remarkably, the predicted θ F-TS2 and θ B-TS2 values nicely agree with the angular positions of the structures located at the largest angles. This holds for all energy sharings considered in this work, from equal (panels (a)) to highly unequal (panels (c)) sharing. We thus conclude that the forward and backward structures positions in Fig. 1-3 are strongly influenced by the TS2 contribution, according to the above qualitative kinematical analysis where the TS2 process is considered as two successive, independent (e,2e) SI of the target. The first-order SO and TS1 contributions might of course be also present. Their contributions should be at their maximum in the momentum transfer direction, θ K , and its opposite, θ -K , but their intensity appears there to be appreciably smaller than that of the TS2, more so for the backward lobe.

It is worthwhile noting that the positions θ F-TS2 and θ B-TS2 predicted by our model are generated from purely kinematical considerations; hence they almost do not depend on the target for a given set of energies (E a , E b , E c ) and fixed scattering angle θ a . The weak target dependence (see last column of Table 1 for a numerical illustration of this point) only shows up via the small change in incident energy (∆E 0 < ~4%) due to the change in SI and DI potentials of the considered targets. The same statement holds for the positions θ K and θ -K which define the momentum transfer axis, and hence the position of the "first-order lobes".

Therefore, within our kinematical model the shift ∆θ b of the predicted lobes from ±K axis is quasi-independent of the target. Even more, it is also quasi-independent of the s or p character of the ionised orbital (we recall that we deal here with two electrons ejected from the 2p, 3p or 3σ g orbital of Ne, Ar and N 2 , respectively). This is indeed the experimental finding, where ∆θ b is practically constant in Fig. 123, for the same (E b :E c ). This gives a kind of "universal", target independent character to the observed shifts and positions of the 4DCS forward and backward lobes.

We may note from Fig. 1-3 that the measured forward and backward angular distributions exhibit more structures than the ones located at about θ F-TS2 and θ B-TS2 and which we interpret as being due to TS2 contribution, or those located at about θ K and θ -K , and which we attribute to SO and/or TS1 contributions. The origin of these additional structures may be diverse [2].

For instance, some of these structures may be due to other higher-order mechanisms not considered here within our kinematical model. A possible such candidate is the Two-Step1-Elastic mechanism (TS1-El) recently invoked by Schulz et al [20] in the context of fast p + He collisions. It can be viewed as follows: first, a TS1 process leads to the ejection of both target electrons, and then the projectile is elastically scattered from one of the two continuum electrons (hence the alternative label '3steps -2 interactions' 3S2).

Also, we want to make two additional comments. First one is related to the ambiguity carried by the (e,3-1e) experiments, for these are not fully differential experiments due to the integration over the undetected 'c'-electron. For DI of He(1s -2 ), its kinetic energy E c is known from the energy conservation E 0 -IP 2+ = E a +E b +E c , where IP 2+ is the He DI potential. But for the more complex targets Ne, Ar and N 2 , the final state of the ion is not determined, so that there is not just one single possible energy for the undetected 'c'-electron, but several discrete E c values in the case of DI. (Triple or higher order ionisation would yield a whole continuum of energies E c but the choice made here of E c = 12 eV, that is E c smaller than (IP 3+ -IP 2+ ) does not allow triple or higher order ionisation to take place). A complete (e,3e)

experiment is needed to resolve this ambiguity. Nevertheless, according to van der Wiel and Wiebes [11], the DI process removing two electrons from the outermost orbital with the ion left in its ground state should largely dominate higher order processes.

Our final comment is about the cross section distributions in Fig. 123. These two-lobes distributions look roughly like (e,2e) SI distributions, which are made of the familiar binary and recoil peaks. It is well established (see e.g. [21]) that, in the incident and ejected energy ranges considered here, the recoil-to-binary (R/B) maximum intensity ratio for the (e,2e) SI case is generally, although not systematically, smaller than unity and is rapidly decreasing with increasing ejected electron energy. In contrast, we observe here for the case of DI a backward-to-forward (B/F) maximum intensity ratio which is more or less constant vs E b .

[We note from Fig. 2 that DI of Ar(3p -2 ) is a special case in this respect, for it shows first a decrease then an increase of the B/F ratio when increasing E b , similar to the peculiar behavior reported for the B/R ratio in the SI of Ar(3s -1 ) [21]]. This different behavior of the B/F and B/R ratios in DI and SI, respectively, gives an additional proof (if at all necessary) of the very fundamental difference in nature between DI and SI. In particular, the recoil lobe in (e,2e) SI has a very different origin from that of the backward lobe in (e,3e) DI, the former being due to a backscattering of the ionised electron from the nucleus, whereas for the latter the intervention of the nucleus is not strictly needed as is the case for instance in our kinematical model.

Conclusion

The previous (e,3-1e) experiments for DI of helium at few hundred eV impact energy are here extended to more complex targets, namely neon, argon and molecular nitrogen.

Compared to He, the ejected electron angular distributions exhibit similar features, that is large shifts from the momentum transfer axis as well as marked structures in the forward and backward lobes. For Ne and Ar, these features are not reproduced by a first Born B1-3C model, nor by a second Born (B2-2CG) model which does only slightly better. At the time of writing there exist neither elaborate first order nor second order calculations to which the present experimental data for N 2 could be compared. The kinematical description of the twostep 2 mechanism developed in [1] is applied to these new data, and it is shown to successfully predict the angular positions of some of the structures of the measured lobes. This work together with the previous one on He thus constitutes a large body of experimental evidence that under the present kinematics the TS2 mechanism dominates over first order SO and TS1 mechanisms. We also observe that the measured angular shifts of the forward and backward lobes are quasi-independent of the nature of the target, as also supported by our kinematical model.

Caption to figures

  nucleus and the third function describes one electron in the field of the second electron, i.e. electron-electron correlation. Still for Ne, this B1-3C model was simplified by replacing the third Coulomb function by the Gamow factor, hence the name B1-2CG[15]. The B1-2CG results differed from those of the B1-3C model only in magnitude, the shape of the angular distributions being essentially unchanged. Consequently, due to the very long computational time needed for the full B1-3C model, it was decided to perform only B1-2CG calculations for the Ar case. For both Ne and Ar targets, we have included the contributions of the three residual states, namely 3 P, 1 D and 1 S. Both these B1 calculations are shown as dashed curves (B1-3C) and dotted curves (B1-2CG) in Fig.1 and 2, where available. We recall here that they only include first-order DI mechanisms, namely the SO and TS1. To account for the second-order TS2 mechanism, calculations were also performed for Ne and Ar within the framework of the second Born approximation where the final state is described by the approximate 2CG wave function (two Coulomb functions plus Gamow factor, hence the name B2-2CG for this model). Here again, the use of the full 3C wave function with the second Born approximation needs much computer time especially for (e,3-1e). The B2-2CG results are shown as full curves in Fig.1 and 2. For N 2 molecule, no similar B1 nor B2 theoretical results do exist. Note that the experiments are obtained on a relative scale and are normalized to the calculations at the maximum of the forward lobe.

Figure 1 (

 1 Figure 1 (Colour online): Relative four-fold differential cross sections (4DCS) for double ionisation of Ne (2p -2 ). The scattered electron with energy E a = 500 eV is detected at an angle θ a = -6°in coincidence with the fast-emitted electron with energy E b , whereas the slowemitted electron with energy E c remains undetected. Panel (a): (E b : E c ) = (12 : 12) eV, (b): (E b : E c ) = (74 : 12) eV, (c) (E b : E c ) = (144 : 12) eV. Full squares are the experimental data, with one standard deviation statistical error bar. Theoretical predictions from the first-Born B1-3C, B1-2CG and the second Born B2-2CG models are shown with the dashed green curve, the dotted black curve, and the full blue curve, respectively. The 4DCS scale shown is arbitrary, where experimental and theoretical results are inter-normalised for best visual fit at the maximum of the forward lobe. The thin dotted vertical lines indicate the direction of the momentum transfer (θ K ) and its opposite (θ -K ). The heavy dashed vertical lines indicate the directions of ejection of the 'b'-electron, (θ F-TS2 and θ B-TS2 ) as predicted by the given kinematical analysis, see text.

Figure 2 (

 2 Figure 2 (Colour online): Same as in figure 1 but for double ionisation of Ar (3p -2 ).

Figure 3 (

 3 Figure 3 (Colour online): Same as in figure 1 but for double ionisation of N 2 (3σ g ) -2 .

Table 1

 1 deg)

		718.6	144	27 / 207	67 / 315
	Ne	646.6	72	36 / 216	77 / 299
		586.6	12	50 / 230	86 / 280
		699.2	144	29 / 209	67 / 318
	Ar	627.2	72	40 / 220	77 / 301
		567.2	12	56 / 236	87 / 278
		699	144	29 / 209	67 / 318
	N 2	627	72	40 / 220	77 / 301
		567	12	56 / 236	87 / 278

. Kinematical parameters considered in this study. The last column indicates the forward and backward directions of ejection of the 'b'-electron (θ F-TS2 and θ B-TS2 , respectively) as predicted by our kinematical model, see text.

Acknowledgment

C Li wishes to acknowledge a PhD grant from the China Scholarship Council.


Guillaume Huillard

Xavier Noblin

Jean Rajchenbach

Propagation of acoustic waves in a one-dimensional array of noncohesive cylinders

Keywords: numbers: 43.25.+y, 45.70.-n, 43.25.Ed, 46.40.Cd Published in Phys. Rev

By means of a photoelastic method, we access the visualization of acoustic waves propagating in a one dimensional array of noncohesive cylinders. As pointed by Nesterenko in the case of spherical grains (V.F. Nesterenko, J. Appl. Mech. Tech. Phys. 24, p. 567 (1983)), the nonlinearity of the contact law between the grains induces a dependence of the wave velocity both on its amplitude and on the confinement force. Our experimental method allows one to access the evolution in time of the internal state of stress of individual grains with an excellent accuracy. We show that the velocity of the sound presents two regimes as a function of the confining force. For low forces, the dependence is strongly nonlinear, while it weakens for higher forces. By means of the direct visualization of the contact zone, we show that both micro-and macroscale imperfections of the surface of contact explain the low forces behavior. We test the consistency of our experimental findings results with both the theoretical expectations and with the experimental determination of the force-displacement dependence. We show moreover that the main damping process originates in solid friction.

I. INTRODUCTION

The disorder in the grain positions, the noncohesiveness of the material, the nonlinearity of the contact law between grains and the interferences between waves passing through different paths render arduous the theoretical treatment of the sound propagation in granular media [1][2][3][4][5][6][7] Nevertheless, understanding the properties of propagation of sound waves in granular media is of crucial interest, e.g., in order to design new acoustic attenuators or, for instance, to improve the interpretation of threedimensional (3D) acoustic tomographic data or seismic recordings.

Since waves are transmitted from one grain to some of its neighbors via the contact network, the nature of the contact plays a crucial role. Due to the nonlinearity of the contact law [START_REF] Landau | Theory of Elasticity[END_REF], the sound velocity depends on the initial state of the (static) pre-stress of each contact. Hence, both the force-displacement relationship between grains in contact and the topography of the contact network have to be accounted for. The second issue then proceeds from the packing disorder. Even if the positions of the grain centroids are ordered in a lattice, some contacts can be open owing to a very small deviation in the particle size or shape and the actual contact network is thus disordered [START_REF] Hermann | [END_REF]. It is well known that an applied external load induces a filamentary network of stress (called force chains) in granular packings, where a modest fraction of the total number of grains carries the main part of the force [10]. Note moreover that the passing of acoustic waves of large amplitude can in turn modify the contact topography. Interestingly, data obtained by Liu and Nagel [2] emphasized the role of interference between acoustic waves following different paths. The complex acoustic signal (the coda) obtained in 3D geometries was interpreted as the signature of a speckle phenomenology, which emphasized the high sensitivity to geometrical rearrangements induced by, e.g., small temperature drift or by the wave propagation itself.

Due to the increase of the contact area with the normal load, the relation between the normal component F 0 of the contact force and the displacement δ 0 between the grain centroids is nonlinear. For spherical beads, this relationship is known as Hertz's law [START_REF] Landau | Theory of Elasticity[END_REF] and reads F 0 ∝ δ 3/2 0 . As a consequence, the wave velocity V increases with the static confining force F 0 . Nesterenko modeled the sound propagation in a one-dimensional nonlinear array [1,[START_REF] Nesterenko | Dynamics of Heterogeneous Materials[END_REF], and he showed that various types of sound waves can propagate through these media, according to the wave amplitude:

(i) Provided that the amplitude of the sound wave is very small compared to the static confining force F 0 , linear acoustic waves can propagate. Theory predicts that the linear wave speed increases as V ∝ F β 0 , with β = 1/6 for spherical beads. Experiments have been carried out in 1D, 2D and 3D for different packing geometries of spherical beads [4,[START_REF] Coste | [END_REF]13].

(ii) At higher amplitudes, when both nonlinearity and dispersion play a role, a cnoïdal wave or a Korteweg-deVries type soliton can be encountered.

(iii) For excitation amplitudes even higher than the confining force F 0 , a new type of excitation (the Nesterenko's solitary wave) is predicted [1]. The speed of this solitary wave varies as the power 1/6 of its amplitude. The existence of such a nonlinear excitations has been recognized experimentally and numerically [1,3,14,15]. The present study is devoted to the propagation of a linear compressive pulse along a 1D array of photoelastic cylinders. Instead of measuring the flight time of the pulse through the array, here we have direct access both to spatial and temporal quantitative informations by means of a photoelastic visualization, which is in real time. The high photoelastic constant of the material allows for an accurate determination of the stress state of each individual grain as a function of time, with a sampling rate up to 10 +5 s -1 . Shukla et al. were the first to use the photoelastic method to probe dynamically wave phenomena, mostly in the non linear regime [16,18]. In a previous paper [START_REF] Noblin | 6th Inter-national Conference on the Micromechanics of Granular Media[END_REF], we have presented a simple quantitative way to measure forces using the photoelastic technique. We focus here on the linear waves speed, and their dissipation.

One of the goals of the present work is to highlight the mechanism which governs the dependence of the wave speed on the imposed static confining force. In some cases, using various materials (steel, brass, glass and nylon spheres), the Hertzian prediction with β = 1/6 has been recovered in 1D arrays [START_REF] Coste | [END_REF]. Nevertheless, deviations from the above expected value has been reported for low confining forces, with experimental determinations of β giving a value close to 1/4, that both in 1D [START_REF] De Billy | [END_REF], 2D [13], and 3D geometry [4,21].

Two reasons have been invoked to explain this discrepancy. The first one relies on the disorder of the packing: some contacts, initially open, are progressively closed when the static confining pressure is increased. Therefore, the macroscopic compressibility of the sample is different at low or high pressure [22]. Another explanation involve a local deviation to sphericity at a microscopic level [START_REF] Goddard | Proc. Roy. Soc[END_REF]. The discrimination between these two different mechanisms has not been elucidated yet. To sum up, the relation between the experimental value of β and the nonlinearity in the contact law remained unclear.

In the present paper we show that the speed increases as a power β ≈ 1/4 of the confining force F 0 , for the low values of the latter. For larger values of F 0 , the exponent β is significantly reduced. Our experiments reveal that these deviations from the expected velocity dependence on F 0 originate in both micro and macroscale imperfections of the contact and subsequent unexpected variations of the surface contact areas with pressure.

Another aim of this work is to study the attenuation of the pulse amplitude while propagating along the array, and to propose a simple model accounting for it.

In the following, we describe first the photoelastic method and the experimental setup (Sec. II A and II B). We then present then experimental results concerning the wave velocity as a function of the static confining force (Sec. III A), the related variation of the mean contact area (Sec. III C) and the force-displacement law (Sec. III B). Next, results concerning the damping processes are shown (Sec. III D). In the last section (Sec. IV), we discuss and propose some elementary explanations capable of accounting for our experimental findings. We use a photoelastic method to obtain quantitative information on the stress state of individual grains in a 1D array. The experimental setup is sketched in Fig. 1.

The system that we studied consists of a linear array of 36 identical cylinders. Grains are prepared by cutting a 9.4 mm-thick polycarbonate plate and machining identical cylinders with diameter d = 2R = 13 mm. The Young's modulus, the Poisson ratio and the mass per unit of volume of polycarbonate are respectively E = 2.76 GPa, ν = 0.38 and ρ = 1, 2 g.cm -3 . Most of the previous studies dealt with spherical grains, while here we use cylinders. Note moreover that the topological disorder proper to 2D or 3D geometries does not intervene in the 1D geometry that we probe here , so that , the acoustic behavior originates merely on the nature of the contacts.

The elastic grains are then packed into contact along a common generatrix, and confined in a 470 mm long duraluminium frame. To ensure the alignment of the cylinders in the parallel direction of their axis, a groove (0.2 mm deep and 10 mm wide) has been machined along the bottom plate of the cell. Moreover, grains can simultaneously undergo an horizontal confining force F 0 and a vertical force F ⊥ . This last one is applied by loading the top lid and prevents the natural buckling of the chain. Both these forces can be varied and measured with precision.

A mechanical pulse generated by means of a tweeter impacts the first grain of the array. A piezoelectric sensor and a static force sensor are positioned after the last grain of the array. They allow access to the static confining force F 0 and the pulse amplitude. They are complementary to direct photoelastic measurements performed in situ . In order to increase the resolution relative to the position of the birefringence fringes, the light is filtered by means of an interferential filter centered around 650 nm. The propagation of the acoustic pulse through the array is then recorded by means of a high speed digital camera (Phantom 7.3) with a resolution of 736 * 32 pixels. Note that a cylinder diameter corresponds typically to 30 pixels. The sequence of images is then processed and analyzed by means of a program written in Matlab language. The bases of the photoelastic method are discussed below.

B. The photoelastic method

Consider a photoelastic cylinder positioned between a couple of identical circular polarizers. Light propagates through grains along the direction parallel to their axis of symmetry. The emerging light intensity is a function of the stress in the plane of the cylinder at each point (x, y) and reads as:

I(x, y) = I 0 (x, y) cos 2 2π λ Ce [σ 1 (x, y) -σ 2 (x, y)] (1)
where I 0 is the incident intensity, λ is the wavelength of the light, C is the photoelastic constant of the sample, e is the thickness of the sample and σ 1 , σ 2 are the local principal stresses at the point (x, y). This relation holds everywhere in the material.

In order to access the state of stress of the individual grains, we compare the light intensity in the central region of each grain with that obtained by means of a calibration procedure. This calibration consists of measuring the transmitted light as a function of the amplitude of an adjustable external uniaxial load F . The central region of measurement is less than 1% of the surface of a grain. The calibration curve is shown in Fig. 2. The intensity measured at the center of the grain, I center , can be expressed as:

I center = (I max -I min ) cos 2 π F ∆F + I min ( 2
)
where F is the compressive force, and ∆F is the force increment corresponding to the passing of a black fringe to the following black fringe. I max is the maximum transmitted intensity and I min is the residual illumination corresponding to the black fringe. In the following, we note:

ζ = (I center -I min ) (I max -I min ) (3)
ζ is the normalized intensity transmitted at the center of the cylinder and takes value between 1 (white fringe) and 0 (black fringe). We can therefore access to the force F by computing:

F = ∆F (n -1) + (-1) n-1 π arccos( ζ) (4)
where n is the fringe order. We measure for our grains ∆F = 42 N. Note that this method is only applicable for forces of the same order on both sides of a grain, with the resulting measurement giving their average value.

C. Procedures

The typical protocol used to conduct measurements is the following. First, particles are positioned in the groove. For a measurement corresponding to a given static confining force F 0 , the vertical force F ⊥ is adjusted in order to ensure the vertical alignment of the array. Then the closest static forces corresponding to black and white fringes located in the grain centers are measured. Hence the values of I max and of I min are deduced for each particles. Note that the determination of I max and I min are required to measure the transient force corresponding to the passing of the pulse. Then a function synthesizer generates a pulse to the tweeter (time-width 100 µs) and a sequence of pictures is recorded with a frame rate 89000 fps. After the tweeter has impacted the first grain, an acoustic pulse propagates through the chain. A sequence of pictures showing the propagation of the pulse is showed in Fig. 3. Note that the spatial extension of the wave is limited to a few grains. Our image analysis program allows one to access the quantity ζ and then the dynamical force F n (t) = F 0 + f n (t) for each step of time and for each particle (where the index n is relative to the grain position in the image. In most cases, the first grain of the image corresponds to the third grain of the chain). In the following, the maximal amplitude of the pulse for the grain n is noted f n,max , i.e.:

f n,max = max [F n (t) -F 0 ] (5)

III. EXPERIMENTAL RESULTS

A. Wave velocity in the linear regime Fig. 4 shows the different curves of F n (t) for the grains numbered n = 8, 13, 18, 23 as a function of the time. Before the pulse emission, all the curves coincide and correspond to the static force F 0 . The measurement of F 0 proceeds from two independent determinations: first, F 0 is measured by means of the photoelastic method, but, we can also access the data issued from the static force sensor. Nevertheless, note that a discrepancy can arise, which originates in the solid friction with the confining boundary walls. As a consequence, when this effect is large, the photoelastic method is preferred. Then, for a given grain, the pulse arises and the force reaches the maximal value before decreasing and oscillating. Note that in this set of experiments, f 1,max is always very small compared to F 0 which ensures the regime to be linear. We observe that the wave propagates through the array, and that the spatial extent of the pulse remains constant during its propagation (within our experimental precision). The spatial width of the pulse is typically 6 -7 grain sizes, so that dispersive effects intrinsic to discrete lattices are negligible along the present chain comprising 36 cylinders. Considering an underestimated wavelength of seven grain sizes, we find that the speed difference between the linear dispersion relation (Eq. 11) and the actual sinusoidal one for the spring-mass chain is about 3 %, which is small. Interestingly, we note that the compressive pulse is followed by a tensile component (the detailed analysis of this phenomenon is beyond the scope of this paper). Finally we note that the amplitude of the pulse decreases uniformly along the chain. The origin of this attenuation will be discussed in-depth in Sec. III D.

From the measurement of the time corresponding to the mid-height pulse amplitude located at the grain of index n, we deduce the distance covered by the pulse as a function of the time (see Fig. 5). This dependence is fairly linear, whatever the confining force. We can de- duce therefore the celerity of the wave as a function of F 0 (see Fig. 6). Clearly, two regimes have to be distinguished. For F 0 < F c ≈ 20 N the wave speed increases rapidly with the static force F 0 . Assuming a power law dependence for the velocity as V ∝ F β 0 , we determine an exponent β ≈ 1/4. This is a clear signature of a highly nonlinear contact law. On the other hand, Fig. 6 shows a significant decrease of the exponent β (β ≈ 1/9) for a larger confining force (F 0 > F c ≈ 20 N ). Actually this regime seems more consistent with the theoretical prediction suited for cylinders, for which the displacementforce relationship is logarithmic (see Sec. IV). In Fig. 6, we have also traced the velocity-confining force curve V (F 0 ) calculated from V = λV with V given by Eq. 11 and k(F 0 ) computed from the experimental static F 0 -δ relation (see Fig. 8). We find good agreement and we interpret the two regimes as a consequence of a change in the contact law for increasing force due to roughness of the contact at microscales and discrepancy with the perfect cylindrical shape at macroscales. Nevertheless, a numerical coefficient has to be used, with λ = 1.48 instead of 1. This discrepancy may be due to the fact that in our frequency range the dynamical stiffness is higher than the static one.

To check the importance of the microscale roughness effect, we have also performed measurement with polished grains (see triangles, Fig. 7). Their microscopic roughness (arithmetic mean value) is about R a 0.37 µm as compared to the non-polished grains for which R a 0.75 µm, the maximum crest to crest distance being respectively 2 µm and 4 µm. The velocity curve presents roughly the same aspect as for unpolished grains, with a lower exponent at large forces. For low forces differences are not significant due to the relative poor precision on F 0 . For high forces, local exponent have been measured as β 1/9 in the non-polished case and β 1/11 in the polished case. The difference in roughness was not FIG. 6: (Color online) Plot, in log-log scale, of the wave velocity V versus the static force F0. For F0 < Fc ≈ 20N, velocity increases rapidly with the static force, we measure an exponent near to 1/4. For F0 > Fc, the dependence of the speed wave with F0 is weaker (exponent 1/9). The solid curve is the theoretical prediction based on Eqs. 10 and 11. The dashed lines are also deduced from Eq. 11 but using the stiffness determined from the experimental force-displacement law (see Fig. 8) and a numerical prefactor λ = 1.48.

significant enough to find strong differences in the forcevelocity relationship, however at high forces a difference in the exponent could be highlighted, expressing the fact that the theoretical behavior for a perfect cylinder was recovered for these grains. The logarithmic contact law for cylinders gives locally an exponent close to 1/11, which is in good agreement with the smooth surface grains that present behavior close to those of the perfect cylinders.

We have also performed measurements by cycling the variation of the confining force: F 0 was first increased and then decreased in order to detect the possible occurrence of hysteretic effects, originating in plastic events at contact. Actually, we did not observe any significant change of velocity up to F 0 100 N.

B. Force-displacement relationship

To access the static force-displacement relation of the array, we simply measured the total displacement of the chain for various F 0 . Once again, precautions have to be taken in order to avoid chain buckling. Therefore, a vertical load is applied to the upper lid to prevent any buckling. Note that the vertical load must be moderate in order to avoid noticeable effects on solid friction. Indeed, noticeable friction between the horizontal grain array and the boundary walls would lead to an significant effective decrease of the confining force along the array (this phenomenon is discussed in depth in Sec. III D. In Fig. 8, We have plotted the relation between the force F 0 and the displacement δ for one grain. One can clearly distinguish two different regimes, consistent with the V -F 0 curve. From these data, we can calculate an effective stiffness, and then using an elementary springmass model (see Section IV) it is then possible to predict the V -F 0 relationship in the studied system (see Fig. 6).

C. Surface of contact between cylinders

In order to have a better understanding of the crossover phenomenon exhibited by both the V -F 0 and F 0 -δ dependences, and to the static force-displacement curve, we have performed a direct visualization of the contact zone between two adjacent cylinders. We access the variation of the contact areas as the static confinement load is increased by means of the following simple method. The array of grains is diagonally lightened up by means of a spotlight. The actual contact area appears as bright (because the light is scattered), while the unloaded regions of the cylinder surface appear as dark (see experimental setup in Fig. 9).

It is therefore straightforward to access the variation of the contact area as the external load is increased. In Fig. 9 a sequence of pictures shows the variation of the lightened actual contact between two adjacent grains as the static confining force is increased for two different families of grains (polished or non-polished). First we notice that a slight imperfection on the cylindrical shape of the grains leads to an incomplete contact along the cylinder generatrix. Second, as the force is increased, the microscopic roughness induces a local increase of the contact area. Using image analysis, we determine the variation of the contact area with F 0 . Results are presented in the Fig. 10. Clearly, for the collection of the polished grains, the behavior expected for ideal cylinders is met as the confining force F 0 is larger than ≈ 20 N. For the case of non-polished grains, the theoretical slope is recovered roughly above 50 N. In both case, the cross over is indicated by an arrow in Fig. 10, it also corresponds to the velocity change as function of F 0 (see Fig. 6). From both micro-and macroscale defects of cylinder surfaces, it results, at low force, a more rapid increase of the contact areas with F 0 , compared to the case of ideal cylinders in contact along a common generatrix.

D. Dissipation

In the following, we address the issue of the dissipation processes which governs the pulse damping during its propagation. Figures 4 and11 clearly show a fairly linear decrease of the pulse amplitude with the covered distance. Interestingly, in Fig. 11, the decay slope does not appear to depend significantly on the initial pulse amplitude f 1,max . Let us define α as the slope of the decay law of the pulse amplitude with distance (expressed in cylinder diameter unit). Hence α (defined in force unit) represents the decrement of the force amplitude when the pulse is transmitted from one grain to its following neighbor. Note that the slope α may depends on the lateral confining static force F 0 and on the vertical load F ⊥ applied on the top lid. These features will be discussed in Sec. IV.

An attenuation length L a (expressed here in grain di- ameter units) can be defined as the distance for which the pulse amplitude vanishes to zero. It is interesting to examine the dependence of the attenuation length L a both on lateral confining force F 0 and on the vertical load F ⊥ applied on the top lid.

In Fig. 12, the variation of the attenuation length L a is plotted as a function of the initial amplitude of the pulse, that for various confining forces (F 0 = 8, 14 or 26 N) and vertical loads (F ⊥ = 0.15, 0.22 or 0.49 N per grain). In all cases, the attenuation length L a is seen to depend linearly on the initial pulse amplitude f 1,max .

This behavior is consistent with Eq. 13. Note that the experimental points ( and ) obtained with the same vertical load F ⊥ (F ⊥ = 0.15 N per grain) and two different values of the lateral confining force (F 0 = 8 and 14 N) merge on the same line. That proves that the damping effect does not vary significantly with the lateral confining pressure F 0 . On the other hand, the set of points displayed in Fig. 12 shows that the attenuation length decreases as the vertical load F ⊥ is increased. As a consequence, α only depends on the vertical force F ⊥ . Next, in Fig. 13, we plot the coefficient α as a function of F ⊥ ; it appears unambiguously that α increases linearly with F ⊥ . This behavior is interpreted in the following section.

IV. MODEL AND DISCUSSION

A. Model

Current models aimed at describing the propagation of acoustic waves through a linear array of grains rely on the analogy with a system of masses and springs [START_REF] Nesterenko | Dynamics of Heterogeneous Materials[END_REF]. For a system of spheres in contact, springs must be considered as nonlinear, to be consistent with Hertz's law, which expresses the relation F 0 ∝ δ 3/2 between the compression force and the displacement δ between the sphere centroids. This last relation [START_REF] Landau | Theory of Elasticity[END_REF] proceeds from the exact equilibrium solution of the linear elasticity equations, and the nonlinearity of the Hertz law results merely from the increase of the area of contact with F 0 . It is worth noting that the contact is unilateral, that is, the contact force becomes zero as soon as the distance between the sphere centroids is larger than the sum of the sphere radii.

For spheres in contact, the contact area is obviously a disk. Let us now consider equal cylinders of length L and radii R. In the following we present the variations of the surface area, displacement and wave velocity for cylinders. Then we propose a simple model accounting for the pulse damping.

Surface area

In the case of two long cylinders (i.e. L >> R) in contact along a common generatrix, the contact area is a rectangle. The surface area S varies with the compressive force as [START_REF] Johnson | Contact mechanics[END_REF]:

S = 16LR E * F (6)
with E * = πE/(1 -ν 2 ).

Displacement

The theoretical relationship between F and δ, F = g(δ), is more complex, and is given approximately, for the displacement (see [START_REF] Johnson | Contact mechanics[END_REF]) by: FIG. 12: (Color online) Distance of attenuation of the pulse given by Eq. ( 13) for different configurations: ( ) F0 = 8N, F ⊥ = 0.15N/grain, ( ) F0 = 14N, F ⊥ = 0.15N/grain, ( ) F0 = 26N, F ⊥ = 0.22N/grain, ( ) F0 = 26N, and F ⊥ = 0.49N/grain. As predicted by Eq. ( 13), La increases linearly with f1,max. The slope is 1/α. It seems to not depend on F0 but decreases when F ⊥ increases.

δ = 2F LE * ln( 4RLE * F ) -1 = g -1 (F ) (7)
Of course, this law holds for ideal cylinders.

Wave velocity

Thus, the system of cylinders maintained into contact by a static compressive load can be viewed as chain of masses M = ρπR 2 L and nonlinear springs obeying Eq. 7. We note u n the displacement of the mass n compared to its equilibrium position. The dynamics of this system is then described by the set of equations:

M ün = g(δ 0 -u n + u n-1 ) -g(δ 0 -u n+1 + u n ) (8)
δ 0 is the displacement of the centroid due to the static force F 0 and depends on F 0 according to Eq. 7 (with δ = δ 0 and F = F 0 ).

In the linear regime |u n -u n-1 | δ 0 , the system (Eq. 8) can be linearized and reads therefore as:

M ün = k (u n+1 + u n-1 -2u n ) ( 9
)
where k represent the stiffness of each contact:

k -1 (F 0 ) = 2 LE * ln 4RLE * F 0 -2 . ( 10
)
The wave equation is linear in this regime but leads to a wave velocity V which depends nonlinearly on the confining force and which reads as:

V = 2R k(F 0 ) M (11)
Note moreover that both the Hertz law and Eq. 7 proceed from the equilibrium solutions of the elasticity equations. Therefore, the analogy with mass and spring systems (as exemplified in Eq. 8 or in Ref. [1]) relies implicitly on the hypothesis that the sound celerity through the material constituting the spheres or the cylinders is very large compared to the pulse velocity through the array.

Dissipation

The linear decay shown in Fig. 11 is consistent with the following equation:

f n,max = -α (F0,F ⊥ ) (n -1) + f 1,max (12)
We attribute this linear decay of the pulse amplitude with distance to the solid friction with the cell walls.

From Eq. 12, we deduce the attenuation length L a (expressed in grain diameter unit):

L a = f 1,max α (F0,F ⊥ ) (13)
On the other hand, it can be deduced from our measurements (see Fig. 13) that α = µF ⊥ . (14) This relation corresponds to the law of Coulomb for the dry friction. A linear fit gives µ = 0.28. This value compares well to the measured coefficient of friction obtained from the onset of sliding of a grain deposited on an inclined plane made of the confining cell material. Thus, we conclude that the main process of dissipation originates in the friction between the cylinders and the slot frame. Such a dissipation mechanism has been highlighted for high amplitude wave propagation in a 1D bead chain by Job et al [START_REF] Job | [END_REF]. In that case the normal force was constant and only due to the grain weight.

Nevertheless, it is worth noting that the attenuation length L a does not exactly tends to zero as the pulse amplitude f 1,max tends to zero (see Fig. 12). It means that there are another sources of damping acting in parallel, but less efficient than the dry friction. This second source of dissipation likely originates from the polymer viscoelestic properties.

B. Discussion

The experimental data displayed in Fig. 6 show that the wave velocity V exhibits two different regimes as the static confining force F 0 is varied. These two different regime can be fitted with a power law: V ∝ F β 0 , with

β ≈ 1/4 if F 0 < F c 1/9 if F 0 > F c .
The cross-over between these two regimes is found for F c ≈ 20 N. In the case of smoother surfaces, the exponent above the critical force is close to 1/11 instead of 1/9, which is closer to the expected theory for ideal cylinders.

For F 0 > F c , the experimental result closely meets the prediction expected from the spring-mass model, with a stiffness consistent with the logarithmic contact law suited for cylinders (Eq. 10).

On the other hand, for F 0 < F c , the dependence of the velocity with the static force is significantly larger than expected from Eq. 7. It results from the fact that, due to micro-and macroscale surface defects, the effective area of contact varies more strongly than expected with the compressive force. Interestingly, we notice a value for the exponent β ≈ 1/4 which compares well with that reported for 1D, 2D and 3D packings of spheres ([4], [13], [START_REF] De Billy | [END_REF], [21]). The exponent seems to be non-significantly sensitive to the packing dimensionality and to the shape of bodies in contact.

In the case of spheres packings, several hypotheses have been advanced to explain the deviation to the V ∝ F 1/6 behavior expected from the Hertz law. Nevertheless, for the linear array studied here, the possibility of contact recruitment (which is a 2D or 3D effect) [26] must certainly be discarded. On the other hand, the hypothesis raised by Goddard [START_REF] Goddard | Proc. Roy. Soc[END_REF] which involves the possible role played by the microscopical surface roughness must certainly be retained. For sufficiently low static force, the contact stiffness is prominently governed by the deformation of asperities, and, e.g., conical asperities would precisely lead to an exponent β = 1/4.

In order to investigate the origin of the exponent β in the law V ∝ F β 0 , we have conducted measurements with two families of cylinders, with two different qualities of surface polishing. These measurements led to close results, except at high forces where an exponent β = 1/9 is found for rough surfaces, and β = 1/11 for smoother surfaces. The last value corresponds very closely to that expected for an array of ideal cylinders.

As shown in Fig. 9, the area of contact between two neighboring cylinders (shown as bright areas in the picture), is not always a rectangle, depending on the quality of the grain surfaces. A explanation for this feature can be that ylinders are compressed along their revolution axis while machined -so that, once freed of the machine, they dilate along the symmetry axis and, in turn, the radius at mid-height undergoes a contraction. This mechanism would thus implies a varying radius along the revolution axis. It might also be envisaged that the axes of the cylinders in contact are not perfectly parallel, but images of the contacts show that this is not the case. We observe that when the confining force F 0 is increased up to F c ≈ 20 N, the contact areas become closer to rectangles and that, in parallel, the theoretical behavior expected for a perfectly-aligned array of ideal cylinders is recovered . Our main conclusion is that imperfection in the perfect cylinder contact, both at micro-and macroscales leads to an exponent β = 1/4 at low forces, and the ideal cylinder case is recovered for high forces.

V. CONCLUSION

We have studied the elastic wave propagation in a 1Dchain of polymeric cylinders. By means of photoelasticity and ultra-high speed visualization, we succeeded in hav-ing a real time access to the local state of stress of individual grains during the passing of an acoustic pulse. We have measured in parallel the variation in the geometry of the contact between neighboring grains as the static confining force F 0 is varied. In the linear regime of propagation, and for pulse widths of the order of six or seven grain sizes, we determined that the pulse velocity varies approximatively as a power law V ∝ F β 0 (with β = 1/4) of the static confining force F 0 , provided that the static force F 0 is smaller than F c ≈ 20 N. Surprisingly, the exponent β = 1/4 compares with that found for packings of spherical grains, and deviates from the theoretical prediction suited for perfect arrays of equal cylinders. Experiments conducted with polished cylinders, or with cylinders of various roughness, lead to small changes at intermediate forces.

In the limit of the large confining forces (F 0 >> F c ), the relation between the pulse velocity and the confining force meets the theoretical prediction. A close look at the contact scale shows that the rectangular geometry of the contact, predicted for perfect cylinders, is recovered in this limit. On the other hand, the geometry of the contact area can deviates significantly from a rectangle for small confining forces.

The effects of a double power law for the contact on nonlinear waves propagation have been studied in [28] and [29]. In these references, the two behaviors come from the special geometry of the objects placed in the chain. In our study we also have a double power law, but due to a multiscale roughness. Another difference is the fact that the nonlinearity decreases with the force.

We have also studied the damping of the compressive pulse as it propagates through the array. We have first shown that the maximum amplitude f n,max decreases linearly with the covered distance. We have also shown that the damping distance is a linear function of the normal load. These two features prove that the main source of dissipation originates in the solid friction with the cell walls. The experimental value of the friction coefficient corresponding to the wave damping matches well with that determined by measuring the onset of sliding of a grain deposited on an inclined wall surface.

FIG. 1 :

 1 FIG. 1: Experimental setup showing 1D grain chains

FIG. 2 :

 2 FIG. 2: (Color online) Photoelastic calibration: we measure the intensity Icenter transmitted through the central region of one grain (white box in (a)) as a function of an applied external uniaxial force F . We deduce from it the force increment ∆F corresponding to the passing of a black fringe to the following black fringe. For cylinders with 13 mm diameter, we measure ∆F = 42 N . (a)-(d): Photoelastic images obtained with the calibration for forces F equal to 0, 10, 20, 40 N .

FIG. 3 :

 3 FIG.3: Example of four successive photoelastic images showing the acoustic pulse propagation. The time between each images is 140 µs and the static confining force is F0 = 4.5 N . The maximum amplitude of the acoustic pulse is f1,max = 10 N and the pulse durationis 100µs. Before the pulse is sent (top image), grains are subjected to the static force F0, which causes the slight fringes near each contact. Then, compressive pulse arises and travels in the chain with a velocity V .

FIG. 4 :

 4 FIG.4:(Color online) Top: Dynamical force Fn(t) vs time for grains n = 8, 13, 18, 23 (from the left to the right) in the linear regime. These curves are the result of the data analysis of photoelastic images by computing Eq. 4 for each grain. Before the pulse arises (f1,max = 1.3 N for the first grain), each grain is submitted to the static force F0 = 10N , then the pulse travels from one grain to another. The temporal width is quite constant for each grain but the maximal amplitude fn,max decreases. Bottom: image sequence from which the curves have been determined. The grain used have been marked, we have divided the image by the first one of the movie, and used false colors.

FIG. 5 :

 5 FIG. 5: (Color online) Distance covered as a function of the time for the wave mid-height of the Fig. 4. The dashed line is a linear fit giving the velocity V .

FIG. 7 :

 7 FIG.7:(Color online) Plot, in log-log scale, of the wave velocity V vs the static force F0 for two surface roughness (triangle: smooth surface, circles: more rough surface). The behaviors at low forces are not representative due to larger errors on F0 in this regime. For F0 > Fc ≈ 20N, the exponent β (V ∝ F β 0 ) is equal to 1/9 for the rough surfaces and 1/11 for smoother surfaces. Straight lines are power fits in the high forces region.

FIG. 9 :

 9 FIG. 9: (Color online) Top: Experimental setup to visualize the contact area. When grains are in contact, they deform themselves and the surface becomes locally plane. Thus, light is transmitted without reflection and pixels are illuminated. In the other case, because of the curvature of the surfaces, light is reflected and pixels appear darker. Bottom : Images of the contact between cylinders for different static forces F0. (a)-(e) Contact surface between two polished cylinders for F0 = 0, 3, 9, 22, 90N . (f) Surface contact at F0 = 90N for non-polished cylinders. We see a clear difference in comparison with (e).

FIG. 10 :

 10 FIG. 10: (Color online) Contact area versus static force F0 for polished cylinders ( ) and non-polished cylinders ( ). The dashed lines are a guide to the eyes, as power law F 1/2 0 which corresponds to the theoretical law for perfect, smooth cylinders (see Eq. 6). The two arrows indicate the cross over forces as discussed in the text.

FIG. 11 :

 11 FIG. 11: (Color online) Plot of the maximum amplitude fn,max vs grain number n. The two sets of data points correspond to different initial pulse amplitude: f1,max = 1.9N ( ) and f1,max = 4.1N ( ) with same values of F0 = 8N and F ⊥ = 0.15N/grain. The amplitude decreases linearly along the chain and the slope of the decrease does not depend on f1,max.

FIG. 13 :

 13 FIG.13:(Color online) Plot of α as a function of F ⊥ . We deduce from these experiments that: α = µF ⊥ , which is analog to the Coulomb law of solid friction. The fit (dashed line) gives µ = 0.28.

Acknowledgments

We would like to thank the referees for their fruitful comments and suggestions on this paper.