Glenn A Hodgkins

Paul H Whitfield

Donald H Burn

Jamie Hannaford

Benjamin Renard

Konrad Stahl

Anne K Fleig

Henrik Madsen

Luis Mediero

Johanna Korhonen

Irstea

Kerstin Stahl

Conor Murphy

Philippe Crochet

Donna Wilson

Climate-driven trends in the

HAL is a

Need for study Need for study

•

  River basins used to study changing floods over time often are influenced by direct human alteration. • Many previous national or regional studies analyzed changes in floods over time, such as the annual peak flow. No studies are known that analyzed changes in major floods (greater than 25-year floods) at a continental scale for minimally altered basins. • To be useful for assessing climate-driven trends, monitored streams need to be relatively free of confounding human influences such as river-basin land-use change (e.g. urbanization), diversions, abstractions, and reservoir regulation (Whitfield et al., 2012; Burn et al., 2012). Results • Analyze flood flows from newly-assembled dataset comprising daily streamflow from minimally altered basins in North America (United States and Canada) and Europe (France, Spain, Switzerland, Germany, United Kingdom, Ireland, Iceland, Norway, Denmark, Sweden, and Finland) • Estimate peak flows with recurrence intervals of 25, 50, and 100 years at each gauge, using consistent methodology: GEV with L-moments (Hosking, 2013) • Compute annual peak flows that exceed these flood thresholds in each of the last 50 to 80 years • Sum the number of annual flood exceedances for continents and other groups: major Köppen-Geiger climate regions (Peel et al., 2007), catchment size, and combinations • Compute trends in the number of large floods with logistic regression (Frei and Schär, 2001) • Glenn Hodgkins: gahodgki@usgs.gov • Burn, D.H., Hannaford, J., Hodgkins, G.A., Whitfield, P.H., Thorne, R., and Marsh, T., 2012, Reference hydrologic networks II. Using reference hydrologic networks to assess climate-driven changes in streamflow, Hydrological Sciences Journal 57, 1580-1593. • Frei, C. and Schär, C., 2001, Detection probability of trends in rare events: theory and application to heavy precipitation in the Alpine region, Journal of Climate 14, 1568-1584. • Hosking, J. R. M., 2013, L-moments. R package, version 2.1. • Peel, M. C., Finlayson, B. L. and McMahon, T. A., 2007, Updated world map of the Köppen-Geiger climate classification, Hydrology and Earth System Sciences 11, 1633-1644. • Whitfield, P.H., Burn, D.H., Hannaford, J., Higgins, H., Hodgkins, G.A., Marsh, T., and Looser, U., 2012, Reference hydrologic networks I. The status and potential future directions of national reference hydrologic networks for detecting trends: Hydrological Sciences Journal 57, 1562-1579.

Contact information and references

Summary of results

Summary of results

1961-2010 1931-2010

The logistic regression odds ratio approximates the change in probability of occurrence between the beginning and end of the period


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  Quantifying the Uncertainty in Discharge Data Using Hydraulic Knowledge and Gaugings with their Uncertainty: the BaRatin approach Benjamin Renard, Jérôme Le Coz, L. Bonnifait, F. Branger, R. Le Boursicaud, I Horner, Valentin Mansanarez, M. Lang, Sylvain Vigneau To cite this version: Benjamin Renard, Jérôme Le Coz, L. Bonnifait, F. Branger, R. Le Boursicaud, et al.. Quantifying the Uncertainty in Discharge Data Using Hydraulic Knowledge and Gaugings with their Uncertainty: the BaRatin approach. AGU Fall Meeting, Dec 2014, San Francisco, United States. pp.1v e e q u a t i o nSoftwarewtool:wBaRatinAGEVGBq-freely-available-3windows-platform2VEBA-currently-under-development:Just-send-an-email-to-baratinBdev@listsBirsteaBfr multi/platform propagation-to-hydrographs improved-graphical-outputs expected-release:-end-of-EAGN more-importRexport-facilities

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Currentkandkfuturekresearch

n c e t a in lim n i g r a p h P r o p aga t e a n


S Besnault
email: sophie.besnault@suez-env.com

S Martin Ruel

J-M Choubert

H Budzinski

Cecile Miege

M Esperanza

N Noyon

S Garnaud

Marina Coquery

Fate of adsorbable micropollutants through sludge drying and composting processes

come

Introduction

Material and methods

• Micropollutants selection and analytical methods

The authors would like to express appreciation for the support of the ONEMA and R+I Alliance, financing this project. We especially thank all technical staff of analytical laboratories and process teams for their contribution.

• Sludge treatment facilities

• Recycling of sludge to land for agriculture : > 50% sludge disposal routes in Europe and USA

• New legislations on reduction of micro-pollutants emissions to the environment :

• EU Water Framework Directive (2000): 33 + 8 substances

• Sewage Sludge Directive (Update of 86/278/EEC, expected)

• Good global performance of biological processes, only 20% priority substances remain in treated wastewater but most of them (60%) expected to accumulate in sludge • Coning and quartering method (ISO 5667-13:1998) Four sludge treatment facilities were selected: two thermal drying facilities, and two composting facilities

•

EC27

Drying : High temperature dryers tend to reduce the concentration in sludge of a higher amount of compounds, but complex phenomena : temperature and sludge retention time (Henry's constant), type of drying (adsorption and adhesion forces), type of sludge, history of dryer

Composting : Most of the analysed substances were detected in the raw sludge (inlet of the composting), hormones (except estrone) are never detected in the compost, some PCBs and PBDEs have lower concentrations in compost than in raw sludge, HAPs have a variable behaviour

  Presence of micro-pollutants in sludge confirmed => Importance to address fate of priority substances in sludge line and identify efficient treatments • Difficulty : sample a representative quantity of the batch to follow Definition of an audit protocol for drying processes • Number of elementary samples and sampling frequency calculated with formulas given in the European standard NF EN ISO 5667-13:1998 • With V the "batch" volume • With Q the volume of the batch, G the maximal flow in the dryer and n sp the number of samples Definition of an audit protocol for composting processes • Number of elementary samples and sampling frequency calculated with formulas given in the European standard NF EN ISO 5667-13:1998


Monitoring of the 2011 batch

1.1 Origin of the batch 1.2 Rearing conditions 1.3 Feeding 1.4 Mortality 1.5 Rearing density 1.6 Growth 2. Simulation of freshwater migration 2.1 Objectives 2.2 Protocol 2.3 Results

Conclusion

Monitoring of the sexual maturation

Technical and management meetings

Résumé

Ce rapport présente les travaux conduits par Irstea, Aquarium La Rochelle et l'équipe Borea dans le cadre de la mise au point d'une méthode d'élevage ex situ de juvéniles de grande alose Alosa alosa et du suivi de la maturation sexuelle. L'augmentation de la biomasse a conduit à l'installation d'un second filtre biologique en avril 2014. La mortalité a été plus forte en 2013 -2014 à cause de pathologies bactériennes et de nodules calcaires dans le rein. Les analyses histologiques des prélèvements effectués en juin 2013 ont montré un développement normal des gonades. Une troisième série de prélèvements a été réalisée en Juin 2014.

Monitoring of the 2011 batch

Origin of the batch

This batch arrived in Aquarium La Rochelle on 24 May and 25 October 2011. These 2 groups were put together in December 2011, with a total number of 602 juveniles. Salinity was increased from 10 to 30 ‰ during the course of December 2011 (a one week transfer).

Rearing conditions

Fish were in a 8 m 3 tank from September 2012 (Figure 1). The net put to prevent fish to jump outside was not totally effective. The system was modified in October 2013. General rearing conditions were not modified (Figure 2). Mean water temperature has been increased from 17.6 ± 0.3 °C during the previous period to 19.3 ± 0.4 °C in order to obtain a better growth. Mean salinity was 30 ± 0.9 ‰, ie two points less from the previous period. Photoperiod was constant at 12/12. Because of the biomass increase, the biological filtration capacity must be increased too. The nitrite concentrations recorded (0.3 mg.l -1 ) proved that the efficiency limit of the biological filter has been reached. A second biological filter system has been put in at the end of April 2014 (Figure 3) with a degasing procedure at the end of June 2014. Current speed was measured in March 2014. At 20 cm from the inner wall and at mid depth, the current was 13.4 ± 0.4 cm.s -1 .

Feeding

Fish were fed with artificial feed (3.5 mm floating pellets) and frozen zooplankton (krill), the same feeding regime as the previous period.

A total amount of 123.63 kg of pellets and 39.90 kg of krill have been distributed during this period. The mean feeding ratio was 1.05 and 0.34 for the artificial feed and zooplankton respectively. A theoretical feeding ratio has been calculated according assessment of intermediate weight gain during the period and food distributed daily (Figure 4). The feeding ratio decreased from December 2013 to prevent degradation of water quality because the biological filtration capacity limit was reached regardless to biomass. Once the additional system was installed, the feeding ratio has been increased. Mortality was higher during this third year of rearing (Figure 5). Monitoring of the survival is only based on natural mortality; fish collected for biological sampling or fish transferred to aquarium exhibition or in freshwater for test, were not counted. Natural mortality corresponded to 95 fish. Two main mortality phases were observed. The first during November -December 2013 correspond to a parasitic and bacterial pathology which was stopped following a 6 days treatment. The second was observed from April. Dead fish were examined and presented calcium nodules in the kidney ("nephrocalcinose") due to too low pH and hardwater (Figure 6). Mortality was stopped thanks to the addition of the new biological filtration and degasing system. At the end of June 2014, 292 fish remained in the tank.

Rearing density

Assessment of the rearing density from January 2012 is plotted in Figure 7. Rearing density reached 4.8 kg.m -3 in March 2014, and was reduced to 4 kg.m -3 at the end of June following mortality and samplings. Following veterinarians' opinions, rearing density in this kind of experimental rearing should not be higher than 2 kg.m -3 to avoid pathologies and to obtain a correct growth. Following this advice, a second rearing tank will be installed in the end of 2014.

Growth

The growth until June 2014 is represented in Figure 8. Specific growth rate (SGR, % body length day -1 ) is 0.07 for length and 0.22 for weight between June 2013 and June 2014.

Food efficiency (or food conversion ratio) is basically assessed from biomass gain and dry food distributed FCR = Food distributed / biomass gain In our case, around 25% of the food distributed is frozen food, ie with a high level humidity, which doesn't allow comparison with aquaculture data. Moreover some fish (86) were removed from the rearing experimentation for samplings or aquarium exhibition which leads to a bias in the calculation. Considering these comments, food conversion ratio calculated during the third year of rearing is 16.3, so 16.3 kg of food (dry and moist food) is necessary to product one kg of fish.

Length -weight relationship was calculated from the beginning of the rearing. The relationship is plotted on Figure 9. Simulation of freshwater migration

Objective

Between 3 and 6 years, adult Shad leave the ocean to enter in the estuary and river to reach the spawning grounds and spawn. During this migration, Shad switch therefore from a salted to a freshwater environment.

In the C1 'ex situ stock' action framework, the aim is to bring to maturity a lot of Shad reared from the larval stage. In order to move closer the biological cycle of the species, it is therefore necessary to simulate the migration of adults to the spawning grounds by transferring the fish in fresh water.

Protocol

The transfer has involved 38 randomly caught Shad. In the case of mortality due to handling, there will be possibility to replace the dead during 7 days after the transfer.

Beyond 10 deaths during the first week, a rapid consultation of the three partners will decide on the continuation of the experimentation of transfer.

The transfer was managed in early April, with 3 year-old fish, with a gradual transfer on 2 weeks. The objective was to maintain the fish for 2 months in fresh water. The fish were transferred into a 3 m 3 tank (2 m diameter).

The decrease of salinity was gradual and continuous, by adding freshwater during the day, to obtain a decrease of approximately 1‰ per day. At the end of the first week of transfer, salinity was around 15‰. During the second week the freshwater input was increased to reach 0‰ at the end of the week.

During the freshwater phase, limited inputs of sea water were achieved (up to maximum 5‰) to prevent possible development of diseases (fungal or bacterial).

The temperature was planned to increase just before the sampling phase (which would correspond to the reproductive phase), with a passage from 20-21-22-23 ° C.

The current speed was maintained around 15 cm.sec -1 . The duration of the light phase was also increased according to the following scheme:

In April: photoperiod 12N / 12L (from 7 h to 19 h) In may: photoperiod 11N / 13L (from 7: 00 to 8: 00 pm) In June: photoperiod 10N / 14L (from 07: 00 to 21: 00)

Feeding was maintained at least at the beginning of the transfer. The batch was observed to detect loss of appetite (fish do not eat during the migration phase). As food intake decreased, feeding was reduced, or even stopped.

Results

With only 4 dead, juvenile shads have well withstood transfer to freshwater (4 more deaths following injuries with an air diffuser). Majority of shads seemed to rapidly stop feeding.

By the end of April, level of NO 2 increased because of the death of bacterial biomass in the biological filter (because of freshwater), which involved the progressive death of all fish.

Conclusion

Three year-old shads can tolerate transfer from seawater to freshwater within two weeks, without significant mortality.

Next year the test will be manage again, with a biological filtration system allowing to maintain good water quality in freshwater.

Monitoring of the sexual maturation (Borea Research Unit)

3.1 Histological analysis of the gonads of two year old shads.

Twenty shads had been sampled in June 2013.

Tissue sampling and histological procedures

Twenty two-year old shads had been sampled during the previous reporting period (June 2013, see 2 nd Report). At two year of age, their body length ranged between 11.7 and 17.9 cm (SL) and their body weight between 21.2 and 77.3 g.

Gonads could be recognized by eye along the body walls, during the body dissection. Ovaries were larger than the testes and with a light orange colour, while testes had a white colour (see Photos, 2nd Report). Gonads were dissected out and quickly placed in freshly prepared Bouin's fixative solution, for histological analyses.

Histological procedure was performed as described in the 2 nd Report.

Histological determination of gonad sex

Histology confirmed the visual sex determination, except for two fish (n° 38 and 42), for which the histological samples collected did not include the gonads. These two fish were removed from the analyses.

Altogether, histological analysis of the gonads could be successfully performed for 18 two-year old shads, and revealed that they corresponded to 9 females and 9 males.

Histological determination of gonadal stage

Females

Histology of the ovaries showed the presence of numerous primary oocytes, characterized by a dense ooplasma, and a large nucleus with many perinuclear nucleoli (for example, see Photos 10 and 11: female n° 40). This indicates that two year-old females are at the primary oocyte stage. As compared to one-year-old females (see 2 nd Report), the oocytes of two-year old females are enlarged, but still at the primary oocyte stage.

Measurement of oocyte diameters

The diameter of the 10 largest primary oocytes was measured for each female. The mean diameter ranged between 93 ± 3.4 (SEM) µm to 137.7 ± 4.9 µm according to fish (Table I). The mean diameter of the largest oocytes of two-year old shads was 109.8 ± 4.3 (SEM) µm.

As compared to one-year old fish (See 2 nd Report: 36.7 ± 4.6 micrometers) these results indicate a 3 fold increase (P< 0.001) in the primary oocyte diameters in two-year old fish, which reflects the progress of primary oocyte growth (Figure 13).

Males

Histological observation of the testes showed numerous spermatogonia, with many figures of mitosis. This indicates that the two-year old males are at a spermatogonial stage with active gonial division. The most advanced stages showed formation of cysts (for example. see Photo 14: male n° 25). As compared to one-year old males (see 2 nd Report), the testes of two-year old males are larger, but still at the spermatogonia proliferation stage.

Biometric parameters of two year old females and males

There was no significant difference between two-year old females and males (Table II) for the body length (females: 14.9 ± 0.6 cm (SEM) ; males: 15.4 ± 0.6 cm; P=0.90) nor for the body weight (females: 47.2 ± 5.3 g; males: 50.6 ± 5.6 g; P=0.90). These parameters reflect the large body growth of the two year-old fish, as compared to one-year old fish (see 2 nd Report) with a mean 2 fold increase in body length and 7 to 8.6 fold increase in body weight for male and female respectively.

The gonadosomatic index (GSI: gonad weight x100/ body weight) ranged between 0.75 and 1.14 in females and between 0.09 and 0.37 in males. Thus, in two-year old shads, the GSI in females (0.94 ± 0.04) was significantly larger than in males (0.17 ± 0.02) (P < 0.0001).

The hepatosomatic index (HIS: liver weight x100/ body weight) was 0.67 ± 0.07 in females versus 0.74 ± 0.05 in males (P= 0.41).

The digestive tract-somatic index (DTSI: digestive tract weight x100/ body weight) was similar in females (6.99 ± 0.49) and males (6.70 ± 0.60) (P=0.82).

Immunoenzymatic assay (Elisa) of 11 Ketotestosterone in adult Alosa alosa

11 Keto-testosterone (11-KT) is an androgen characterized in teleosts, and used in some species as a marker of sexually mature males. However, 11-KT was shown to be secreted not only by males but also by females in some species, such as the eel, Anguilla species. We investigated the presence of 11-KT in the plasma of adult Alosa alosa.

Samples

Plasma were sampled from 5 males and 1 female wild adult Alosa alosa caught at Migado's field station, Bruch (June 14, 2012), for inducing reproduction (Table III). Plasma samples were stored at -20°C until assay. Several tests were performed with serial doses of Alosa plasma samples. Dose-response curves, parallel to the 11-KT standard, could be obtained between 10 and 0.2 microliters of plasma (Figure 15). In the 5 males, plasma 11-KT levels ranged between 164 and 542 pg/ml (Table 3) (mean value = 321 ± 69 pg/ml). The highest level was found in the male with the highest GSI.

11-KT was also well detectable in the plasma of the female, with a value of 136 pg/ml.

Validation of 11-KT assay after plasma extraction

In order to assess whether the detection of 11-KT in female Alosa was not an artefact due to crude plasma sample, extraction of the plasma was performed by SB. 100 microliters of plasma were mixed with 200 microliters Elisa Buffer and 3 ml Ether and then frozen in liquid nitrogen (12 sec). The Ether phase was collected, evaporated at 37 °C. Samples were recovered in 200 microliters of Elisa buffer. 11-KT could still be detected by Elisa in extracted plasma samples of both male and female Alosa alosa.

These results indicate that 11-KT is produced by both mature males and females in Alosa alosa and thus that this androgen is not specific of the males. Such a production by both males and females is also the case in some other teleosts, such as the eel, and other fish such as the sturgeon.

Preliminary Elisa tests for other steroids

Multiple preliminary tests were performed by SB on unextracted and extracted Alosa alosa plasma samples for immunoassay of estradiol. However, we found that the kits available at that period (Labor Diagnostica Nord GmbH, Ref FR E-2000) were unsuitable for estradiol assays in fish plasma, due to artefactual crossreactions. Further tests were performed using unextracted and extracted eel plasma, and new kits were tested and compared. Finally the kit Nova Tec, Ref DNOV003 was selected, based on results in the eel.

For Testosterone, we have selected and validated the kit Eurobio Ref EUDNOV002, based on eel plasma samples.

These estradiol and testosterone Elisas are now ready for validation in Alosa alosa. We need more plasma samples from wild adults Alosa alosa in order to perform these validations.

Samplings carried out in 2014

Samplings took place the 16 and 17 June 2014.

Twenty two shads were sampled after euthanasia in a concentrated bath of eugenol.

One fish (n°47) was the last survivor of the transfer to freshwater performed during the spring. This male fish had the smallest size, likely reflecting high stress and arrest of feeding. All the other fish were from the batch maintained in seawater.

The same protocol was followed as for the previous series of sampling. Fish were individually anesthetized with clove oil. The following external biometric measurements were performed: body length, body weight, body height (at the level of the pectoral fin). After a ventral incision, blood was punctered from the heart using heparinized syringes. Plasma collected after centrifugation was kept frozen at -20°C.

Gonads were easily observable by eye in three-year old fishes (Figures 16), so that they could be dissected out, weighed (for the determination of gonadosomatic index, GSI = % gonad weight/body weight). Gonad was fixed in freshly made Bouin's solution for histological analysis.

The liver and digestive tract were also dissected and weighed for the determination of the hepatosomatic index (HSO) and digestive tract-somatic index (DTSI), respectively.

As such fishes are very unique and precious, additional tissues were sampled, out of the scope of the present project, in order to anticipate possible future molecular investigations: brain and pituitary were dissected under Optika binocular and stored in RNAlater solution (Ambion) at -20 °C. Gonad, gill and liver were also sampled and stored in RNAlater. Some kidney samples, with visible pathological nodules, were fixed for histology. During the dissection, ovaries (larger size and orange colour) could be distinguished from testis (smaller size and white colour). Histological analyses to be performed during the next reporting period will allow to assess these observations.

Based on these preliminary observations (to be confirmed), among the 22 fish sampled, 13 would be females and 7 males, with a doubt for two individuals (to be assessed by histology). They would not differ by their body length (21.0 ± 0.6 cm for females versus 18.1 ± 1.3 for males) nor by their body weight (129.5 ± 13.9 g for females versus 93.0 ± 22.5 g for males).

Females had a significantly higher their gonadosomatic index than males (GSI 0.99 ± 0.08 % in females versus 0.23 ± 0.08 % in males; P=0.0038).

Three-year old females Alosa alosa also had a significantly higher hepatosomatic index than males (HSI 1.09 ± 0.14 in females versus 0.72 ± 0.12 in males; P=0.0323). This may reflect a higher liver metabolism in females possibly related to vitellogenesis.

Action D7

A new batch of 27 shads has been transferred in the aquaria exposition in September 2013.

A new informative panel will be installed in 2015. This panel will group together information concerning three migratory fish presented in this area, shad (Alosa alosa), sturgeon (Acipenser sturio) and eel (Anguilla Anguilla). An illustration of the project is presented in Figure 17.

3. 1

 1 Histological analysis of the gonads of two year old shads 3.2 Histological determination of gonadal stage 3.3Biometric parameters of two year old females and males 3.4 Immunoenzymatic assays of 11 Ketotestosterone in adults 3.5 Samplings carried out in 2014 3. Aquarium exhibition (Action D7)

Figure 1 :

 1 Figure 1: The 8 m 3 rearing tank with the new net to prevent escapements.

Figure 2 :

 2 Figure 2: Monitoring of the temperature and the salinity from July 2013 to June 2014

Figure 3 :

 3 Figure 3: The biological filtration system installed in April 2014.

Figure 4 :

 4 Figure 4: Theoretical feeding ratio during the period 2013-2014

Figure 5 :

 5 Figure 5: Survival after 3 years of rearing

Figure 6 :

 6 Figure 6: Presence of calcium nodules in the kidney (arrows)

Figure 7 :

 7 Figure 7: Rearing density from January 2012 to June 2014

Figure 8 :Figure 9 :

 89 Figure 8: Linear and weight growth until June 2014

Figure 10 :

 10 Figure 10: Histological section of ovary from two-year old female Alosa alosa (Female n°40; x 4) -Numerous primary oocytes are observed.

Figure 11 :

 11 Figure 11: Histological section of ovary from two-year old female Alosa alosa (Female n°40; x 20) Primary oocytes with dense ooplasma. and perinuclear nucleoli are observed.

Figure 12 :

 12 Figure 12: Histological section of ovary from two-year old female Alosa alosa (Female n°31; x 20) Primary oocytes with dense ooplasma and a few lipid vesicles, and perinuclear nucleoli are observed.

Figure 13 :

 13 Figure 13: Oocyte diameter (µm) plotted against length (Ls in cm)

Figure 14 :

 14 Figure 14: Histological section of testis from two-year old male Alosa alosa (Male n° 25; x 40) -Numerous spermatogonia organized in cysts, with many mitoses are observed.

Figure 15 :

 15 Figure 15: Dose-response curves of 11-KT standard and of serial dilutions of Alosa alosa plasma samples in the Elisa for 11-KT

Figure 16 :

 16 Figure 16: Length measurement, A -blood sample, B -testis, C -ovary, D

Figure 17 :

 17 Figure 17: Project of the information panel

Table I :

 I Results of oocyte diameter measurements

		Mean diameter µm	Standard deviation (SD)	Standard error (SE)	Length (Ls) cm
	N°26	108.3	15.3	4.8	16.5
	N°29	111.0	13.9	4.4	14.3
	N°31	113.0	10.9	3.5	15.3
	N°32	96.6	9.0	2.8	14.7
	N°34	105.9	11.7	3.7	11.7
	N°35	96.6	9.0	2.8	17.9
	N°37	116.6	10.6	3.3	15.8
	N°39	93.4	10.8	3.4	14
	N°40	105.4	9.9	3.1	14.3

Table II :

 II Sex and biometric parameters of two-year old Alosa alosa, sampled in June 2013 at the Aquarium of La Rochelle. F: female: M: male; GSI: Gonadosomatic index; HSI: hepatosomatic index; DTSI: Digestive tract-somatic index (DTSI).

	Gonad sex (ovaries or testis) was determined by visual observation during dissection and further
	assessed by gonad histology, except for (?) (gonad not retrieved in the tissue sample collected for
	histology)
	Ref fish	Gender	Body lenght (Ls in cm)	Body weight (g)	GSI	HSI	DTSI
		M	15.3	46.55	0.37	0.60	8.61
		F	16.5	64.09	0.94	0.41	8.88
		M	16.5	54.02	0.11	0.57	9.38
		M	12.2	23.92	0.21	0.96	10.58
		F	14.3	35.42	1.10	0.65	5.98
		M	14	37.63	0.11	0.56	5.87
		F	15.3	54.78	0.86	0.44	8.56
		F	14.7	43.08	0.91	0.93	8.7
		M	15.3	44.52	0.09	0.63	6.45
		F	11.7	21.21	0.99	0.52	6.6
		F	17.9	74.16	0.98	0.94	4.5
		M	17.2	77.33	0.19	0.97	6.97
		F	15.8	52.19	0.75	0.63	6.57
		M ?	16.4	61.45	0.11?	1.01	5.22
		F	14	37.01	0.84	0.54	5.89
		F	14.3	43.06	1.14	0.97	7.24
		M	14.6	40.75	0.10	0.73	6.01
		M ?	12.7	24.29	0.08?	0.66	4.65
		M	17.2	70.43	0.21	0.82	5.56
		M	16.7	59.95	0.1	0.66	4.33

Table III :

 III Biometric parameters and plasma 11-KT levels of wild adults Alosa alosa sampled in June 2012

	Adult	Gender Body	Body	GSI	11KT
	Alosa		length (cm)	weight (g)		(pg/ml)
	1	M	53	1050	4.28	197.69
	2	M	47.5	685	3.64	396.90
	3	F	65	2405	8.52	136.03
	4	M	55	1190	6.3	542.55
	5	M	49.5	985	2.03	164.81
	6	M	46	540	3.7	305.57
	3.4.2 Assay of 11-KT in plasma samples from male and female adult Alosa alosa
	Elisa for 11-KT were performed by SB, using Elisa Kits (Cayman Chemical Company, Ref 582751)
	previously validated for measuring 11-KT in the eel (Anguilla Anguilla) plasma.

Technical and management meeting

A half day Life+ project meeting was held at Aquarium La Rochelle on 16 June 2014, with Borea Team and Pierre Morinière from Aquarium La Rochelle. Members of Borea Team presented results of the histological analysis performed on the 2013 samples. Discussions focused also on the schedule for the next season.


D Moncoulon
email: dmoncoulon@ccr.fr

D Labat

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The Eqs. ( 1), (2), and (3) in this paper contain an error. The whole equation appears twice in all three cases.

Please find below the correct versions of the equations.

r = p -etp, (1)
δh δt = r -i + q in -q out (-riv), (2)
δθ δt = i -l + u in -u out , (3)
Published by Copernicus Publications on behalf of the European Geosciences Union.


Henri Gouin

Sergey Gavrilyuk

Wetting Problem For Multi-Component Fluid Mixtures

Keywords: Wetting problem; fluid mixtures PACS: : 68, 45, G

In this paper we propose an extension of the Cahn method [1] to binary mixtures and study the problem of wetting near a two-phase critical point without any assumption on the form of intermolecular potentials. A comparison between Cahn's method and later works by Sullivan [2,3], Evans et al [4,5] is made. By using an expression of the energy of interaction between solid surface and liquids proposed recently by Gouin [6], we obtain the equations of density profiles and the boundary conditions on a solid surface. In the case of a convex free-energy, a one-dimensional solution of a linear problem is proposed for the density profiles between a bulk and on a solid wall. A non-linear model of binary mixtures [7] extending Cahn's results for simple fluids is also studied. For the case of a purely attractive wall we have established a criterion of a first order transition in terms of the structure of the level set of the homogeneous part of the free energy. Additively, explicit expressions of density profiles near the wall are proposed. They allow one to consider the adsorption of mixture components by a solid wall.

Introduction

In 1977, Cahn [1] gave simple illuminating arguments to describe the interaction between solids and liquids. His model was based on a generalized van Email addresses: henri.gouin@univ-cezanne.fr (Henri Gouin), sergey.gavrilyuk@polytech.univ-mrs.fr (Sergey Gavrilyuk).

Preprint submitted to Physica A, Volume 268, Issues 3-4, Pages 291-308 (1999).

der Waals theory of fluids treated as attracting hard spheres [7]. It entailed assigning to the solid surface an energy that was a functional of the liquid density "at the surface". Three hypotheses are implicit in Cahn's picture for simple fluids: (i) In order for the liquid density to be a smooth function of the distance from the solid surface, that surface is assumed to be flat on the scale of molecular sizes and the correlation length is assumed to be greater than intermolecular distances (this is the case, for example, when the temperature T is not far from the critical temperature T c ).

(ii) The forces between solid and liquid are of short range and can be described simply by adding a special energy at the solid surface. (iii) The fluid is considered in the framework of a mean field theory. This means, in particular that the free energy of the fluid is a classical so-called "square-gradient functional".

After Cahn, the problem of adsorption and wetting was studied by a statistical method by Sullivan [2,3], Evans et al [4,5], respectively for gas and binary fluid mixtures. From the point of view of Sullivan and Evans et al one may view Cahn's approach as open to criticism for several reasons: (a) Cahn's treatment is based on phenomenological "square-gradient" version of van der Waals theory, which in contrast to the approach initiated by van Kampen [8] does not attempt to relate directly the properties of the nonuniform fluid to the interactions occurring on a molecular level. (b) The density adjacent to the wall vary strongly over the range of intermolecular forces, consequently the gradient expansion approximation used in deriving the square-gradient theory is no longer valid. (c) Cahn's theory leaves unspecified a contribution due to the fluid-solid interfacial free energy.

Evans et al [4,5] following Sullivan's approach [2,3] for simple fluids consider the special case of a contact between a two-component mixture near "the critical end point" and a wall. They used Sullivan's grand potential to describe the solid-fluid and fluid-fluid interactions and tried to solve directly the problem of repartition of densities in a liquid (gas). Evans et al obtain a coupled system of integral equations for chemical potentials (cf. Eq. ( 6) in [4]). Then, to solve the system, it is necessary to know the interaction potentials between components and between solid wall and components: Evans et al assume an exponential interaction both for component-component and solid-components (as in [2,3]). Only such a hypothesis allows one to obtain two differential equations instead of the two integral equations (Eq. ( 10) in [4]). This assumption cannot be obviously valid for large classes of mixtures. Moreover, a special hypothesis (mixing rule) concerning interactions between components is assumed. Then, the mixing rule and exponential dependence allow one to obtain both the linear relation between potentials and boundary conditions and the problem is reduced to the problem of the contact of one-component fluid with a wall.

The phenomenological "square-gradient" model is proposed in case of an infinite non-homogeneous fluid or a fluid mixture as a small-gradient approximation by Widom [9] and Fleming et al [10]. The method is extended in mean-field theory for semi-infinite media in contact with a wall: as proved in [6], the fact that the densities are discontinuous at the solid wall does not disqualify the procedure used by Widom and Fleming et al and Cahn's treatment is valid for fluids and fluid mixtures near a critical point in contact with a wall.

In this paper, we use the expression of a surface energy. The surface is assumed to be solid and interactions between solid and fluids are sufficiently shortrange. The contribution of fluids is represented by a surface free energy with a density of the form E S (ρ 1S , ρ 2S ), where ρ 1S and ρ 2S are the limiting densities of the fluid components at the surface. The expression of the surface energy obtained in [6] is in the form:

E S = -γ 11 ρ 1S -γ 21 ρ 2S + 1 2 (γ 12 ρ 2 1S + γ 22 ρ 2 2S + 2γ 32 ρ 1S ρ 2S ) . (1)
This expression represents first terms of a more complex expansion. It is an extension with explicit calculations of the widely known expression due to Nakanishi and Fisher [11] and examined in a review paper by de Gennes [12].

All the coefficients γ ij can be calculated explicitly after the particular form of interaction potentials was chosen. For example, in the case of London forces, the values of coefficients related to the densities of the two fluids at the surface are [6]

γ 11 = µ 1 π 12 δ 2 1 ρ 3 , γ 21 = µ 2 π 12 δ 2 2 ρ 3 , γ 12 = k 1 π 12 δ 2 1 , γ 22 = k 2 π 12 δ 2 2 , (2)
γ 32 = k 3 π 24 1 δ 2 1 + 1 δ 2 2 ,
where ρ 3 is the density of the solid, µ i , i ∈ {1, 2} are the coefficients associated with intermolecular potentials of interaction between the fluids and the solid wall, k i , i ∈ {1, 2, 3} are intermolecular potentials of interaction between the molecules of fluid i and themselves or between the two fluids and δ i = 1 2 (σ i +τ ), i ∈ {1, 2} are the minimal distances between the solid and molecules of the two species of the mixture, where σ i , i ∈ {1, 2} is the diameter of molecule of fluid i and τ for the solid. Expression (1) allows us to estimate the influence of a solid wall on each component of a fluid mixture. Depending on the values of coefficients γ ij , one can estimate the magnitude of the attraction or repulsion effects due to the wall. As our approach is also based on a mean-field approximation, we assume that variations of densities near the wall take into account several molecular ranges. Hence, it is possible to present the total free energy of the system "fluids -wall" as the sum of a bulk free energy and a surface energy which is an additional contribution arising from the non-uniformity of the fluid near the wall. By using an extended variational principle, we obtain two boundary conditions at the wall and two partial differential equations for the density profiles of the components between a solid wall and a bulk. The complete set of boundary conditions and equations for densities allow us to obtain the profiles of densities in the following physical situations. The first is the study of the linear problem associated with the equilibrium of a two-component onephase mixture near a critical point with a solid wall. The second is the study of the non-linear problem of the contact between a two-component two-phase mixture near a critical point and a wall. We get a condition of wetting and a first order wetting transition in terms of the level set of the homogeneous part of the free energy.

To clarify the presentation some calculations are situated in Appendices. In Appendix 1, we present general calculations by using an extended variational principle applied to multi-component mixtures. In Appendix 2 we give an analytical representation of the profiles of densities connecting bulk and solid wall for a general form of the free energy of a two-component mixture near any critical point. These representations may be used to investigate the adsorption of fluid components of a mixture by a solid wall.

Equations of density profiles and boundary conditions: general results

The general form of the free energy per unit volume of the mixture is proposed in the form [7,10,13]

E = E(ρ 1 , ρ 2 , ∇ρ 1 , ∇ρ 2 ) , (3)
where ∇ notes the gradient operator in the physical space D. The associated total free energy is

E D = D E dD.
The wall boundary S of D is endowed with a surface energy per unit area.

The surface is solid and sharp on an atomic scale and the interactions between surface and fluids are sufficiently short range; the general form of the surface free energy per unit area used is

E S = E S (ρ 1S , ρ 2S ). (4)
Consequently, the free energy of S is

E S = S E S dS.
Then, the grand potential of the system "fluid mixture -wall" is

E = D E dD + S E S dS.
The condition of extremum of the energy E based on hypotheses (3) and ( 4) yields (see for details Appendix 1):

-Equations of two profiles of component densities:

∇ ∂E ∂ρ α -div ∂E ∂(∇ρ α ) = 0 α = 1, 2 , (5)
where

∂E ∂(∇ρ α )
is the vector whose components are the partial derivatives of E with respect to the components of ∇ρ α and div is the divergence operator.

-Two boundary conditions at the solid wall :

n ∂E ∂(∇ρ α ) + ∂E S ∂ρ α = 0, α = 1, 2, ( 6
)
where n is the external unit normal vector to D.

Equations of equilibrium ( 5) are the same as in [7] given for the one-dimensional case. Conditions (6) generalize those proposed in [1].

3 The dynamical system associated with one-dimensional density profiles

In the simplest case, the surface energy per unit area is given by (1) where the coefficients γ ij are expressed by means of a mean-field approximation through the potentials of the intermolecular interactions (see for example (2)) and the free energy per unit volume is of the form

E = U(ρ 1 , ρ 2 ) + 1 2 C 1 (∇ρ 1 ) 2 + 2D ∇ρ 1 ∇ρ 2 + C 2 (∇ρ 2 ) 2 , ( 7
)
where U(ρ 1 , ρ 2 ) is the homogeneous free energy per unit volume and C 1 , C 2 , D are constants such that the corresponding quadratic form is positive definite (we denote the free energy by U corresponding in [7] to -W ).

Let us consider the case of a flat plate wall defined by equation z = 0 (see figure 1), where z denotes the one-dimensional coordinate orthogonal to the wall. The equations of equilibrium (6) associated with (7) are

         C 1 d 2 ρ 1 dz 2 + D d 2 ρ 2 dz 2 = ∂U ∂ρ 1 (ρ 1 , ρ 2 ) + e 1 D d 2 ρ 1 dz 2 + C 2 d 2 ρ 2 dz 2 = ∂U ∂ρ 2 (ρ 1 , ρ 2 ) + e 2 (8)
where e 1 and e 2 are two constants of integration.

These equations are complemented by the boundary conditions ( 6) at z = 0. By using expression (1) of the surface energy, we get

       C 1 dρ 1 dz + D dρ 2 dz = -γ 11 + γ 12 ρ 1 + γ 32 ρ 2 D dρ 1 dz + C 2 dρ 2 dz = -γ 21 + γ 32 ρ 1 + γ 22 ρ 2 (9)
We have to add the condition in the bulk (at z = +∞):

ρ 1 = ρ 1∞ , ρ 2 = ρ 2∞ . ( 10
)
4 Linear wetting problem

We consider the case of a one-phase mixture (liquid or gas) in contact with a solid wall. The densities of the two-components and the temperature are close to critical conditions. Moreover, we assume that density variations are small enough with respect to bulk densities, i.e.

ρ i -ρ i∞ ρ i∞ << 1, i = 1, 2
such that we can consider a linearized problem associated with equations ( 8).

Let us denote

r =    ρ 1 -ρ 1∞ ρ 2 -ρ 2∞    , q = -    γ 11 γ 21    + Γ    ρ 1∞ ρ 2∞    ≡ -γ + Γρ ∞ , Γ =    γ 12 γ 32 γ 32 γ 22    , A =    C 1 D D C 2    , B =     ∂ 2 U ∂ρ 2 1 ∂ 2 U ∂ρ 1 ∂ρ 2 ∂ 2 U ∂ρ 1 ∂ρ 2 ∂ 2 U ∂ρ 2 2     . (11)
The matrix B is calculated in the bulk (ρ 1∞ , ρ 2∞ ). Taking into account the definitions (11), we get the linearized problem associated with equations ( 8)-( 10) in the form:

A d 2 r dz 2 = Br (12)
A dr dz = q + Γr at z = 0 (13)

r = 0 at z = +∞. ( 14
)
The stability of the thermodynamic state of the bulk requires that the symmetric matrix B is also positive definite.

Let χ 2 i , h i be the eigenvalues and the eigenvectors of the equation

B -χ 2 i A h i = 0.
Since B and A are symmetric and positive definite, χ 2 i are positive. We can always suppose that1 h i Ah j = 0, i = j .

The solution of (12) satisfying the condition ( 14) is in the form

r = 2 i=1 b i h i exp(-χ i z) where χ i > 0 . (15)
Substituting expression (15) into condition (13), we get a linear system of algebraic equations for the unknown coefficients b i

2 i=1 b i (Γ + χ i A)h i = -q
which defines a unique solution b i if det(h i Γh j +χ i h i Ah j ) = 0. In particular, if Γ is negligible (we assume that the wall is purely attractive), we get q =γ (see ( 11)) and

b i = γ h i χ i (h i Ah i )
.

In such a case, the solutions satisfy conditions (13) and the density profiles fulfil the solution of linearized problem. Equations ( 15) yield different forms of density profiles. Depending on wall conditions, we may obtain both monotonic and non-monotonic profiles. This is similar to results of [7] in the non-linear case without a solid wall. In figure 2, we represent the different density profiles for each component of the mixture. We note that only one extremum point may appear for each density profile. This result is different from the results of Evans et al where density profiles are essentially monotonic. 5 Wetting problem near a critical point for a two-component mixture

The dynamical system

In a two-phase region near a critical point at a given temperature T , the expression of the free energy per unit volume U associated with a phase equilibrium is of the form [7]

U = a 2 0 (b 2 0 x 2 + y) 2 + (y + t) 2 . ( 16
)
The parameter t is an independently varied field characterizing the "distance" from the critical point (ρ 1c , ρ 2c ), a 0 and b 0 are functions of the temperature.

The variables x and y are defined through the transformation

r = ∆R, R =    x y    , ∆ =    a b c d    , r = ρ -ρ c , ρ =    ρ 1 ρ 2    , ρ c =    ρ 1c ρ 2c    . (17)
The scalars a, b, c, d associated with the physical properties of the mixture near the critical point depend on the temperature T . The constants of integration e 1 = 0 and e 2 = t are already incorporated in U. With Eq. ( 17), the system (8) can be rewritten in the form

∆ ⋆ A∆ d 2 R dz 2 = ∇ R U, ( 18
)
where ∆ ⋆ denotes the transpose matrix and ∇ R U means the gradient with respect to R.

Following Rowlinson and Widom [7] we denote by

M = ∆ ⋆ A∆ =    m xx m xy m xy m yy    . (19) Obviously, if A is positive definite, M is also positive definite, i.e. m xx > 0, m xx m yy -m 2 xy > 0.
The boundary conditions ( 9) at the wall are

M dR dz = g + G R, where g = ∆ ⋆ (-γ + Γρ c ) and G = ∆ ⋆ Γ∆ . ( 20
)
In the following, we choose in Eq. ( 16) a 0 = 1/ √ 2 and b 0 = 1 (to do this, we have only to change the values of coefficients of the matrix ∆ defined by ( 17)).

Hence, U = 1 2 (x 2 + y) 2 + (y + t) 2 .

The system ( 18)-( 19) yields

       m xx d 2 x dz 2 + m xy d 2 y dz 2 = 2x(x 2 + y) m xy d 2 x dz 2 + m yy d 2 y dz 2 = x 2 + 2y + t (21)
System ( 21) admits the first integral

1 2 m xx ( dx dz ) 2 + m xy ( dx dz )( dy dz ) + 1 2 m yy ( dy dz ) 2 -U(x, y) = 0. ( 22
)
This integral is similar to the integral of energy for mechanical problems.

Substitution of boundary conditions (20) into the relation ( 22) yields necessary conditions for x, y at the solid wall. For simplicity, we consider only the case of an attractive wall (G is then negligeable). Conditions (20) yield

M dR dz = g. ( 23
)
In fact, it is natural to expect that the results we obtain in the case of an attractive wall are closely similar to the results associated with the most general case. Relations ( 22) and ( 23) yield

U(x, y) = k 2 ,
where k 2 = gM -1 g. Then, discussion of the wetting of a fluid mixture with a solid wall arises naturally from the drawing of the level curves of U(x,y) as a function of the parameter t.

Connection between the dynamical system and Young's conditions

For a solid wall in contact with phases α and β, the contact angle θ is defined with the help of surface free energies σ along the solid surface (Young's conditions)

σ αβ cosθ = σ αS -σ βS .
(24) where the different subscripts designate phases adjoining the surface or interface. No value of θ satisfies Eq. ( 24) unless

σ αβ > | σ αS -σ βS | . ( 25
)
If the inequality (25) is not satisfied, one of the fluid phases completely wets the solid and there is no contact between solid and other fluid phase. In fact, the forbidden surface is replaced by a layer of the wetting phase and the surface free energy becomes the sum of two surfaces' free energies of the layer

σ αS = σ αβ + σ βS . (26)
Condition ( 26) corresponds to the perfect wetting with the solid. The surface energies can be calculated by the formulas

σ αβ = +∞ -∞ (K + U)dz, σ αS = +∞ 0 (K + U)dz, σ βS = +∞ 0 (K + U)dz, where K = 1 2 m xx ( dx dz ) 2 + m xy ( dx dz )( dy dz )+ 1 2 m yy ( dy dz
) 2 and integrals are taken on different paths connecting phase α and phase β or a phase and the wall [7].

Let us note (ds

) 2 = 1 2 m xx (dx) 2 + m xy dxdy + 1 2 m yy (dy) 2 .
From the first integral ( 22), we get

σ αβ = (x β ,y β ) (xα,yα) (2U(x, y)) 1 2 ds, σ αS = (xα,yα) (x Mα ,y Mα ) (2U(x, y)) 1 2 ds, σ βS = (x β ,y β ) (x M β ,y M β ) (2U(x, y)) 1 2 ds . (27)
The integrals ( 27) are calculated on the paths associated with system (21) and the boundary conditions on the wall

x = x Mα , y = y Mα or x = x M β , y = y M β ,
and in the bulks

x = x α , y = y α or x = x β , y = y β .

Discussion of the wetting

For a solid wall the value of k 2 is given. Hence, the discussion depends on the relative value of parameter t.

(a) t > 0 and large enough.

In this case we are far enough from the critical conditions. In figure 3a the phases are in points A(x α , y α ) and B(x β , y β ). One obtains easily that

x α = - √ t, y α = -t, x β = √ t, y β = -t.
The points M α and M β belong to two different connected components of the level set U = k 2 . In the vicinity of A (or B), the energy U is a convex function of x, y and as in Section 4, it is possible to find the profiles of densities connecting A and M α or B and M β , respectively. The integrals ( 27) are positive and σ α,β is large with respect to σ α,S and σ β,S . Then the relations σ α,S ≤ σ α,β + σ β,S and σ β,S ≤ σ α,β + σ α,S

hold and we are in the case of partial wetting with θ = 0.

(b) t > 0 and small enough.

This case corresponds to phases close enough to the critical point (see figure 3b). The level set U = k 2 consists only of one connected component containing the points M α and M β . The phases are at the points A(x α , y α ) and B(x β , y β ). They are very close with respect to the distance to the level curve. The superficial tension σ α,β is small with respect to the free energies σ α,S and σ β,S . The values of σ α,S and σ β,S are in general different and one of the two relations ( 28) is not satisfied. We are in the case where one of the two phases wets completely the solid wall. No contact appears between the other phase and the solid. For exemple, if relation ( 26) is satisfied, the phase β wets completely the wall.

(c) t < 0

The mixture has only one phase at the point A(0, -t/2), which is the only singular point of the system (21). The energy U attains a minimum at the point A (we note that for t > 0 this point corresponds to a saddle point, which is not associated with a bulk phase). The free energy U of the mixture is convex at the vicinity of A (figures 3c and 3d). If t is small enough, the linear solution for the profiles of densities obtained in Section 4 can be used. When t is large enough, the solution for the profiles of densities can be calculated analogously as in the Appendix 2.

Some remarks on the profiles of densities

The system (21) yields and admits the first integral ( 22):

M d 2 R dz 2 =    2x(x 2 + y) x 2 + 2y + t    (29)
dR dz M dR dz -(x 2 + y) 2 -(y + t) 2 = 0. ( 30
)
When the densities x, y are far from critical conditions, t is negligeable with respect to x and y and (30) reads

dR dz M dR dz -(x 2 + y) 2 -y 2 = 0. ( 31
)
Let us denote V (z) = RMR. Then, by using ( 29) and (31), we get d 2 V dz 2 = 3x 4 + 5x 2 y + 3y 2 . The right-hand side is a positive definite quadratic form, which implies that

d 2 V dz 2 > 0 . Hence, V (z) ≥ V (0) + V ′ (0)z .
If dV dz z=0 > 0 , it follows from here that V → ∞ as z → ∞. The level curves of V are represented in figure 4a. Hence, x or y must be an increasing function of z near the solid wall. For example, let x be an increasing function of z when z is small enough. Due to the fact that x → ± √ t as z → +∞ and t is small with respect to x(0), the representation of x as a function of z has the form shown in figure 4b. Hence, the function x(z) is nonmonotonic. In this case we may have also non-monotonic profiles of densities unlike in the treatment of Evans et al. Then, construction of an analytical solution may be done according to the algorithm proposed in Appendix 2.

Conclusion

Near critical conditions, by using a variational approach, we have obtained for an isothermal binary mixture in contact with a solid wall equations of equilibrium and boundary conditions which generalize those obtained by Cahn. With limit conditions in the bulk, they form a closed boundary value problem. When the free energy of the mixture is a quadratic form with respect to the densities of components and their gradients, we get explicit profiles of the densities in the one-dimensional case.

In the case of a purely attractive wall we have also established a criterion of a first order transition, when a contact angle against a solid wall becomes zero. This criterion is formulated in terms of the level set of the function U(x, y): U(x, y) = k 2 , where k 2 depends on the boundary conditions. If the level set is a connected set, two multi-component layers exist: one layer with ordinary adsorption and the second one in contact with the wetting layer. If the level set is disconnected we have partial wetting. We have also shown that the profiles of density are typically non-monotonic. This is in agreement with Rowlinson and Widom [7] where infinite two-phase two-component mixtures where considered.

The variations (A3) are related to the virtual displacements (A1) by the formulae [14,15] δ α ρ α = -div(ρ α ζ α ) , α = 1, 2.

Fig. 1 .

 1 Fig. 1. One-dimensional contact of a fluid with a wall.

Fig. 2 .

 2 Fig. 2. In (a) and (b) the profiles of densities for a component are monotonic. In (c) and (d) the profiles admit an extremum of density for a component. The extremum is not on the wall but at some distance into the bulk measured on a molecular scale.

Fig. 3 .

 3 Fig. 3. Drawing of level curves for the free energy U for different values of t. Points A (and B) correspond to the bulks. The bold curves are paths connecting a phase and the solid wall or two phases. The interfacial tension is calculated along these paths. (a) is the case of partial wetting with a non-zero Young angle. The other figures are associated with different cases of total wetting of one phase: in (b) with two phases and in (c) and (d) with one phase.

Fig. 4 .

 4 Fig. 4. (a) represents level sets of V. Since x or y must be an increasing function of z near the wall, due to the limit conditions at infinity, x or y is a non-monotonic function of z. (b) is a representation of such a function.

  α ρ α + f α δ α ∇ρ α with f α = ∂E ∂(∇ρ α ) . Since δ α ∇ρ α = ∇(δ α ρ α ) we get δ α E D = D δE δρ α δ α ρ α + div (f α δ α ρ α ) dD, div (ρ α ζ α )dS .The variations of the surface free energy areδ α E S = S ∂E S ∂ρ α δ α ρ α dS = -S ∂E S ∂ρ α div(ρ α ζ α )dS .The grand potential of the system is E = E D + E S and its αth variation is given by the formulaδ α ζ α )dS . Denoting c α = n f α + ∂E S ∂ρ α , we obtain S c α div(ρ α ζ α )dS = S c α div S (ρ α ζ α ) + c α n ∂(ρ α ζ α ) ∂x n dS ,

VN W ≡ V ⊗ (N W) denotes the bilinear form of vectors V and W with respect to matrix N ; the bilinear form is symmetric when matrix N is symmetric.

Acknowledgements

We have greatly benefited from comments and advises of Professor Benjamin Widom.

Appendix 1. Calculus of variations for fluid mixtures

We study a two-fluid equilibrium, but the method can be extended to any number of components. The position of a two-fluid mixture is associated with two applications

where X α denote the Lagrangian coordinates belonging to a reference space D α associated with the αth component and x denotes the Eulerian coordinates in the physical space D [13]. The virtual motions of particles are deduced from the relation

Here ε α , α = 1, 2 are small parameters defined in a neighbourhood of zero.

Virtual displacement ζ α are defined by [13,14]

At the solid boundary, the virtual displacement ζ α is subject to the conditions

where n is the unit normal vector to the boundary.

Eulerian variations of densities are defined by

and δ α ρ β = 0, β = α, α, β = 1, 2 . (A3)