Ni Zhu
email: ni.zhu@univ-eiffel.fr

Miguel Ortiz

Valérie Renaudin

Cécile Ichard

Sander Ricou
email: sander.ricou@sgme.com

Dataset of the intermediate competition in Challenge MALIN: Indoor-outdoor inertial navigation system data for pedestrian and vehicle with high accuracy references in a context of firefighter scenario

Keywords: Pedestrian navigation, inertial sensors, GNSS, indoor & outdoor positioning, Infra-Red stereo Specifications Table

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Subject

Signal Processing in multisensory positioning

Specific subject area

Indoor-outdoor pedestrian and vehicle navigation with multiple sensor fusion techniques -Ublox-M8N GNSS receiver: GPS and GLONASS data; -Realsense D435i: tri-axis accelerometer, tri-axis gyroscope, compressed infrared stereo images, compressed depth image; -Xsens MTi-300: tri-axis accelerometer, tri-axis gyroscope, tri-axis magnetometer, barometer, thermometer; -PERSY: tri-axis accelerometer, tri-axis gyroscope, tri-axis magnetometer, GNSS receiver: GPS and GLONASS data

Type of data

How data were acquired

The data were collected with several sensors (inertial sensors and GNSS receivers) attached to different parts of the subject's body. The scenario is designed to simulate the firefighters' mobility during an intervention.

Data format

Raw data: Raw GNSS data; Raw inertial and magnetic data PERSY Calibrated inertial data Xsens Analyzed/Filtered data: the final position estimates computed by the CyborgLOC team during the competition.

Parameters for data collection

The activities include walking, running, stair-climbing, side-walking, crawling, passing above/below obstacles, carrying a stretcher, ladder climbing, etc. The travelled environments include indoor, outdoor, urban canyons. The indoor part contains different stair levels: from the underground up to the 6th floor.

Description of data collection

The data was collected in the framework of the indoor/outdoor positioning competition MALIN, which aims at designing a real-time positioning system for the military or the first response emergency services.

Data source location

City/Town/Region: Blois Country: France

Data accessibility

Repository name: Zenodo Data identification number: 10.5281/zenodo.3785045 Direct URL to data: https://doi.org/10.5281/zenodo.4290789

Value of the Data

 This multiple sensor dataset is collected during an indoor/outdoor positioning competition in a challenging scenario of firefighter intervention. The sensors are mounted on different parts of the subject's body. The high accurate ground truth is partly provided, which allows evaluating the performance of positioning algorithms.

 The following people may benefit from this dataset: Researchers who work in the domain of pedestrian positioning, data scientists who are interested in human activity classification or pattern recognition, and biomechanics researchers working on human gait analysis, etc.

 The performances of positioning algorithms applied to this dataset could be a benchmark of the current existing positioning techniques. The scenario design methodology is also useful for future experiment design.

 The various travelled environments and complex activities of the subject make the dataset full of challenges for positioning. The maps and evaluation metrics are also provided, which allows different users to assess their positioning algorithms.

1. DATA DESCRIPTION

Experimental setup

As shown in Fig. 1, the experimental setup includes the following sensors: one Ublox NEO-M8N GNSS receiver located on the right shoulder of the subject, a Realsense D435i stereo camera mounted on the left shoulder of the subject, three Xsens MTi-300 fixed respectively on the right shoulder, under the sternum and on the left foot of the subject and one PEdestrian Reference System (PERSY) on the right foot of the subject. Table 1 provides the technical specifications of the different sensors including the sampling frequency and the measurement units.

EXPERIMENTAL DESIGN, MATERIAL AND METHODS

Scenarios

The two scenarios took place respectively on the 25 th (Evaluation scenario) and the 26 th (Reference scenario) September 2019, at the Fire and Rescue Training Center (CFIS41) in Vineuil, France to simulate the situation of firefighters during interventions. As shown in Fig. 2, the site includes buildings with stairwells, corridors, rooms of different surfaces (from 3 to 50 squared meters), and floor heights with/without windows, basement, etc. The indoor path goes through several floors (from the underground up to the 6 th floor) with different changing floors means such as circular stairs, ladders and inclined planes.

Different motion modes and postures exist in each scenario. They correspond to the typical activities of firefighters such as lying down, crouching, crawling, jumping, sidestepping, carrying a load on the shoulders, crouching (1 knee on the ground) while pulling a load, evacuating a person by stretcher (pair), passing over obstacles of different heights, opening/closing solid/glass doors. Furthermore, environmental difficulties were also set up by the organizers to test the robustness of the localization systems involved in the competition. For example, they consist of walls with various or/and repeated patterns, uniform walls, strong luminosity, quick dimming of brightness, low brightness, mirrors, windows, Cold smoke, moving elements in the field of view (people, vehicles, objects, ...). During the data acquisition, the subject wore a firefighter uniform with rubber firefighting boots (without metal shells) equipped with all experimental sensors mentioned above. The subject carried an axe during the majority of the scenarios.  Preparation: The subject is equipped with the prototype and the whole system is turned on. Two calibration procedures are carried out for PERSY: a dynamic calibration for the magnetometer [START_REF] Renaudin | Complete triaxis magnetometer calibration in the magnetic domain[END_REF] and a static calibration to estimate the subject-dependent parameters to detect the stance phase. These parameters are provided in the "2_Sensors_calibration_and_configuration" file, detailed in section 2.3.

 Vehicle: As shown in Fig. 3, a typical firefighter vehicle is used to transport the subject from the preparation room to the test site. The subject gets into the vehicle and sits on a designated seat, which may differ from one scenario to another. The orientation of the seat relative to the longitudinal axis of the vehicle is unknown. The vehicle moves at a limited speed (< 30km/h) over a distance of approximately 400 m. During this phase, the prototype is activated and then connected to a GNSS Real Time Kinematic (RTK) receiver for possible initialization. The lever arm between the two systems is not provided. When the vehicle arrives at the disembarkation point, the prototype is disconnected from the GNSS RTK receiver and the subject gets off the vehicle.

 Outdoor building approach: The subject walks about 150 m with different gaits and postures.  Known position: The subject stops on a known point marked at the entrance of the building with a fixed orientation. A new initialization of the prototype is possible via a simulated GNSS receiver under the same conditions as the "Vehicle" phase. The simulated GNSS signal can be available or not depending on the entry point on the scenario.  Indoor 2: Same as indoor 1.

 End: The scenario ends outdoors stationary on a marked waypoint.

Sensors reading

The raw data of all the sensors in the prototype is stored in the bag files of the folder "1_Sensor_readings" of each scenario in the Zenodo repository. For each scenario, two bag files can be found: the first one (i.e., X_scenario_data_camera_only.bag1 ) includes only the camera data and the second one (i.e., X_scenario_data_wo_camera.bag) includes all other sensors' data mentioned in Table 1 except the camera.

In the bag files, the raw data are saved in different topics of ROS messages. The list of these ROS topics as well as their full descriptions and example values are given in the annex file "Annex_2_RosTopic_description.pdf" in the subfolder "1_Sensor_readings" of each scenario folder. Several non-standard ROS messages were created by the CyborgLOC team to store measurements non existing in the standard ROS message library. This last information is also provided in the same folder as the bag files with the name "non_standard_ros_message".

To facilitate the use of our dataset under Rosbag format, a toolkit of python scripts named MALIN Data Processing Tools is provided on GitHub (https://github.com/4ggroup/malin_data_processing_tools). It allows merging Rosbags, converting Rosbag files to CSV files as well as republishing camera's topics as decompressed data. Details about these processing tools could be found in the Readme file on the Github page.

Sensors calibration and configuration

The prototype's inertial sensors need to be calibrated to estimate and mitigate the stochastic and deterministic errors. The calibration parameters of each inertial sensor as well as the initial parameters are stored in the folder "2_Sensors_calibration_and_configuration". The following files are included:

-PERSY calibration and initial parameters are in the file "X_scenario_persy_calibration_data.bag2 ", which is detailed in Table 3.

Table 3. Description of PERSY calibration and initialization parameters

Parameters Description Magnetic Field

The reference magnetic field at the test site on the competition day calculated using the most recent World Magnetic Model (WMM)

Allan Variance parameters for accelerometer, gyroscope and magnetometer

Stochastic error parameters obtained by an Allan Variance

Analysis with 14 hours of static acquisition at 160 Hz. Random walk component and bias instability component are provided.

MagCalib

The output results of the dynamic calibration procedure using the algorithm in [START_REF] Renaudin | Complete triaxis magnetometer calibration in the magnetic domain[END_REF] to calibrate the magnetometer.

Thresholds (QSF, QSA, ZUPT)

The outputs of static calibration, which depend on the subject and are used to update the state vector in the Extended Kalman Filter (EKF) during the quasi-static phase of a walking gait cycle and quasi-static magnetic field periods. The algorithm's details can be found in [START_REF] Scornec | Foot-mounted pedestrian navigation reference with tightly coupled GNSS carrier phases, inertial and magnetic data[END_REF].

Qbn

The initial attitude angles of PERSY in the quaternion form. Details about the conversion between Euler angles and the quaternion can be found in [START_REF] Scornec | Foot-mounted pedestrian navigation reference with tightly coupled GNSS carrier phases, inertial and magnetic data[END_REF].

Raw data to calculate the above parameters are in the following files3 : o X_scenario_persy_dynamic_calibration_raw_data.bag o X_scenario_persy_static_calibration_raw_data.bag Except for the stochastic error parameters mentioned above, the determinist error parameters for PERSY accelerometer are also provided in the file "PERSY_calib_determinist_error.txt", which aims at correcting axis misalignment and offset.

-Realsense D435i camera calibration parameters are in the following files, which are detailed in Table 4.

o "X_scenario_camera_calibration_data_allan.yaml" o "X_scenario_camera_calibration_data_calibr.yaml". The imagers we use are the two infrared imagers. They are cameras with a wide angle of view, however data provided are rectified video streams. Therefore, we model our camera with the pinhole model and estimate the intrinsic parameters of the two cameras and the matrix estimating the rotation/translation between these two reference frames thanks to a target called Aprilgrid [START_REF] Olson | AprilTag: A robust and flexible visual fiducial system[END_REF] and the method proposed in [START_REF] Maye | Self-supervised calibration for robotic systems[END_REF].

Extrinsic parameters between IMU and Cameras (imu_T_cam0 and imu_T_cam1)

These parameters provide spatial transformation information between the imagers and the inertial unit, as well as the time difference between the two types of sensors, estimated by the maximum likelihood method in continuous time [START_REF] Furgale | Unified temporal and spatial calibration for multi-sensor systems[END_REF][START_REF] Furgale | Continuous-time batch estimation using temporal basis functions[END_REF].

The measurements of Xsens sensors are already calibrated so the calibration parameters are not provided. Configuration files for Realsense D435i and Ublox M8N GNSS receiver are also provided in the sub-folder "2_Sensor_calibration_and_configuration/configuration". Complementary information or parameters can be found if needed.

Ground Truth

The organizers of the challenge MALIN have estimated reference coordinates (stationary points) and reference volumes (enclosing volumes) to assess the performances of the competing positioning technologies. These reference coordinates considered as the ground truth are also provided in the dataset, in the folder "3_Ground_truth" under each scenario folder.

The reference points and volumes were estimated with a Differential GNSS (DGNSS) post-processing coupled with theodolite records. Assisted by differential corrections provided by permanent base stations belonging to the French networks RGP, DGNSS receivers were used to estimate accurate absolute reference Points of Interest (POI) outdoors (on the ground or the roof of the buildings). Once the maps of buildings were correctly positioned, the next step consisted of surveying indoor POIs using a theodolite, which is an accurate optical measurement system measuring distances and angles between its position to a POI where a target-mirror is setup. Starting outdoors, on a known reference coordinates point, this process constructs step-by-step all needed indoor POI coordinates. The positioning error is controlled with a loop closure error computed on the same starting/ending surveying point.

Accurate geo-referenced coordinates (5-10 cm) with Mean Sea Level (MSL) altitude associated with a predefined reference orientation are provided for each reference stationary point. During the test scenario, as shown in Fig. 4, the subject stood still on each reference stationary point while facing an indicated direction. Both feet of the subject were joined together aligned along the indicated direction to avoid potential misalignment for the foot-mounted sensors.

An enclosing volume is a bounded and geo-referenced volume, such as a cylinder-type one (Fig. 5) or a room-type one (Fig. 6). During the test scenario, the subject performed different motions and specific trajectory patterns within a particular enclosing volume. It corresponds to moving in a bounded 3D space. The georeferenced maps of each floor (from -1 to 6) provided by the organizers are included in the sub-folder "maps" under the "3_Ground_truth" folder. These maps can be combined with the trajectories in kml format using the xml file "BASE_PLAN_CFIS41.xml". To synchronize the ground truth data with the Coordinated Universal Time (UTC) timeframe, the UTC timestamps of the start and end instants of each stationary point are stored in the text file under the "Ground Truth" folder. Similarly, the entry and exist instants for each enclosing volume are given in UTC timeframe. The detailed descriptions of the .xml file and the text file can be found in the annex file "Annex_3_Ground_truth_file_description.pdf" in the "3_Ground_truth" folder.

Videos of the difficulties for each scenario are provided in the sub-folder "video". These videos have been captured by the organizers using cameras that are synchronized at the same timeframe as the evaluation system.

EVALUATION METRICS

Proposed evaluation metrics

In the context of the competition MALIN, the organizers have used specific evaluation metrics to evaluate the positioning performances of each competitor's localization system. For the convenience of the organizers, all the teams were required to provide the estimated positioning information at 5Hz through a unique PLOC format defined in the annex file "Annex_4_PLOC_description.pdf" under the subfolder "4_CyborgLocTeam_output" for each scenario.

The measurements used for the evaluation are the PLOCs stored between the entry and exit instants of each ground truth stationary point or enclosing volume. In the case of missing measurements, there is no interpolation and the errors are set at a constant level of 50m for the horizontal position errors, 15m for the vertical error and 20° for the orientation error. When evaluating the performance inside the enclosing volumes, a compensation of +1m is applied to the altitude calculated by each team's localization system to correspond to localization at the level of the subject's waist since each team was required to provide the altitude at the foot level of the subject.

The location accuracy score is determined based on two criteria: geometric and topological.



Location accuracy scoring -geometric criterion

Geometric location errors are evaluated at the reference stationary points. The 2D geometric error (absolute horizontal errors) and vertical error are determined according to the following definitions.

The score of the 2D horizontal geometric criterion is determined by

𝑁𝑜𝑡𝑒 2𝑑 = 130 𝑛𝑠 • ∑ exp (- 𝜀 𝑖 2𝐷 75% 3 ) 𝑖=𝑛𝑠 𝑖=1
𝜀 𝑖 2𝐷 75% represents 75% horizontal positioning error at i th reference stationary point in meter. ns represents the number of stationary points planned in the scenario.

The score for the vertical geometric criterion is determined by

𝑁𝑜𝑡𝑒 𝑉 = 70 𝑛𝑠 • ∑ exp (- |𝜀| 𝑖 𝑣 75% 3 ) 𝑖=𝑛𝑠 𝑖=1
|𝜀| 𝑖 75% represents 75% vertical positioning error at i th reference stationary point in meters. ns represents the number of stationary points planned in the scenario.



Location accuracy scoring -topological criterion

Topological location errors are measured from the enclosing volumes provided in the ground truth file as well as the timestamp recorded by the organization staff when the subject passes by. It is based on the number of location data provided by the team's system contained in each enclosing volume between the times of entry and exit.

The score for the topological criterion is determined by

𝑁𝑜𝑡𝑒 𝑇 = 200 • ∑ 𝑁 𝑝𝑚 𝑖 𝑉 𝑖=1 ∑ 𝑁 𝑝𝑡 𝑖 𝑉 𝑖=1
𝑁 𝑝𝑚 𝑖 is the number of real measurements provided by the location system contained in the i th enclosing volume for the duration provided by the organization. 𝑁 𝑝𝑡 𝑖 is the number of theoretical measurements contained in the i th enclosing volume at 5 Hz for the duration provided by the organization staff. 𝑉 is the total number of enclosing volumes for the scenario.



Orientation accuracy scoring At each reference stationary point, the subject is required to orient himself toward a predefined direction with the help of ground markers. The orientations provided by the location system are compared with the reference orientations. To calculate a score for orientations, the average absolute errors using all orientation estimates are calculated in degrees (between 0° and 360°). Then, the score for the orientation criterion is determined by 𝑛 is the number of total PLOC outputs and |𝜀| is the absolute value of the orientation's error in degrees.

A python script handy tool is provided on GitHub (https://github.com/4g-group/ploc_evaluation) to evaluate the localization results (in PLOC format) against the ground truth using the metrics mentioned above.

CyborgLOC Team output

The output of the CyborgLOC team is provided in the sub-folder "4_CyborgLocTeam_output" under each scenario folder. It includes a bag file with all the PLOC output, the position errors (e.g. Fig. 7) and the orientation errors (e.g. Fig. 8) calculated using the ground truth as well as the estimated trajectories in kml format at each floor level. Fig. 9 shows the estimated trajectory at the level 0 by plotting the kml file with Google Earth.

CyborgLOC's positioning system is mainly based on the PERSY, which is a foot-mounted positioning device developed by GEOLOC Laboratory at the Gustave Eiffel University. Regarding the positioning algorithm used in PERSY, classical strap-down mechanization of inertial signals is applied while walking gait cycle characterization has been introduced to assist the computation with updates that occur during specific phases of the gait [START_REF] Scornec | Foot-mounted pedestrian navigation reference with tightly coupled GNSS carrier phases, inertial and magnetic data[END_REF]. For example, during the stance phase of a walking gait cycle, zero velocity updates or zero angular rate updates are applied to mitigate the cumulative errors induced by the inertial sensors. During the competition, the GNSS signals from the Ublox NEO M8N receiver was employed in favorable tracking conditions to assist PERSY's positioning algorithm. This is realized by estimating the transformation parameters with a least-square algorithm between two synchronized point patterns, i.e., the GNSS trajectory in favorable tracking conditions and the PERSY trajectory [START_REF] Umeyama | Least-squares estimation of transformation parameters between two point patterns[END_REF].

Fig. 1 .

 1 Fig. 1. Test setup for the acquisition of the data for the indoor competition challenge MALIN

Fig. 2 .

 2 Fig. 2. An overview of the experimental site: the Fire and Rescue Training Center (CFIS41) for firefighters in Vineuil, France

Fig. 3 . Indoor 1 :

 31 Fig. 3. The subject is getting on the vehicle

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Test scenario at a reference stationary point: The subject is with the red helmet looking at an indicated direction Fig. 5. Cylinder-type enclosing volume where the test person remained stationary

Fig. 7 .

 7 Fig. 7. Position errors of the CyborgLOC Team output for Evaluation scenario

Fig. 8 .

 8 Fig. 8. Orientation errors of the CyborgLOC Team output for Evaluation scenario

Fig. 9 .

 9 Fig. 9. Trajectory estimated by CyborgLOC Team (Source: Google Earth)

Table ,

 , Figure, Map, Video, Rosbag, from which the following sensor data can be extracted:

  1.1 General description and structure of the repositoryThe dataset contains inertial signals, infrared images, depth images, Global Navigation Satellites System (GNSS) measurements, pressure, temperature and magnetometer measurements that were collected during the annual competition of the indoor positioning challenge MALIN (MAîtrise de la Localisation INdoor). The same subject, equipped with the test setup, collected the data in two

	using the ROS standard message "Header", including
	a timestamp divided into secs and nsecs. Data are listed by topics in the bag file. A conversion tool
	(https://github.com/4g-group/malin_data_processing_tools) is provided. It allows merging Rosbags as
	well as converting Rosbag files into Common Separate Vector (CSV) files.
	The complete dataset is shared via the Zenodo repository
	(https://doi.org/10.5281/zenodo.4290789). The data is sorted by scenario into two folders, i.e.,
	"E_scanario_data" and "R_scenario_data", containing the data collected respectively during the
	Evaluation scenario and the Reference scenario. The structure inside each scenario folder is the
	same. It includes the following sub-folder: 1_Sensors_readings,
	2_Sensors_calibration_and_configuration, 3_Ground_truth, 4_CyborgLocTeam_output. The content
	of each sub-folder is detailed in Section II: Experimental Scenarios of this paper.

different scenarios: a 1500 m outdoor/indoor reference scenario and 1300 m indoor/outdoor evaluation scenario. The dataset includes also the ground truth of stationary points and enclosing volumes computed by the competition organizers, i.e., the Directorate General of Armaments (DGA) and the French National Research Agency (ANR). The final outputs estimated by the CyborgLoc competition team are also provided.

The different sensors, embedded in the prototype, are coupled and coordinated by the Robot Operating System (ROS) via its powerful communication infrastructures (e.g.: service, message, topics, etc.). The data format is the common ROS data/message type, which is recorded in bag files using the "rosbag" tool. All data samples are timestamped

Table 1 .

 1 Technical specifications of the sensors used for building the dataset

Hardware Type Embedded Sensors Units Dynamic Range Sampling Frequency (Hz) Ublox NEO-M8N GPS

		-GLONASS (L1) receiver	N/A	N/A
	Realsense D435i tri-axis accelerometer	m/sec²	+/-40 m/sec²
		tri-axis gyroscope	rad/sec	+/-1000 °/sec
		IR stereo depth module	N/A	N/A
	Xsens MTi-300	tri-axis accelerometer	m/sec² (calibrated)	+/-50 m/sec²
		tri-axis gyroscope	deg/sec (calibrated)	+/-450 °/sec
		Tri-axis magnetometer	a.u.*	+/-80 uT
		Barometer	hPa	300-11000 hPa
		Thermometer	°C	N/A
	PERSY	tri-axis accelerometer	m/sec²	+/-10 g
		tri-axis gyroscope	rad/sec	+/-400 °/sec
		tri-axis magnetometer	mGauss	+/-8.1 Gauss
		GPS-GLONASS (L1) receiver	N/A	N/A

*The output of the Xsens Magnetometer is in arbitrary units (a.u.), one a.u. is the magnetic field strength during calibration at xsens' calibration lab. This is approximately 40 uTesla (0.4 Gauss).

Table 2

 2

lists the 7 different phases presented in each scenario. The difficulties included in each phase are detailed in "Annex_1_X_scenario_description.pdf" inside each scenario folder.

Table 2 .

 2 Chronological list of the seven phases of the experimental scenario

	Phase	1	2	3	4	5	6	7	8
	Number
	Phase	Prepara-	Vehicle	Outdoor	Known	Indoor 1	Outdoor	Indoor 2	End
	Type	tion		building	position
				approach
	Duration	10 min	2-3 min	2-3 min	20 sec	15 min	1.5 min	15 min	3 min

Table 4 .

 4 Description of Realsense D435i calibration parameters

	Parameters	Description
	Allan Variance parameters for	Stochastic error parameters obtained by an Allan Variance
	accelerometer, gyroscope	Analysis with 12 hours of static acquisition at 400Hz. Random
		walk component and bias instability component are provided.
	Intrinsic parameters for Infra-red
	cameras parameters (fx, fy, cx
	and cy)

« X » represents one of the scenarios, which could be « E » for Evaluation scenario or « R » for Reference scenario.

« X » represents one of the scenarios, which could be « E » for Evaluation scenario or « R » for Reference scenario.

Due to an unexpected situation of the evaluation scenario, the dynamic camlibration for the magnetometer was not realised during the prepare phase. The parameters used are the ones estimated the day before. Thus no raw data of dynamic calibration is provided in the dataset for the evaluation scenario.

Acknowledgment

This research is financially supported by the Directorate General of Armaments (DGA: Direction Générale de l'Armement) and the French National Research Agency (ANR) in the framework of the Challenge MAîtrise de la Localisation INdoor (MALIN) [ANR-17-MALN-0002].

.org/10.5281/zenodo.4290789.

Ethics Statements

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

-Informed consent: Informed consent was obtained from all individual participants involved in the study.


Nigel Clarke

Nigel Gibbions

Didier R Long

Diffusio-osmosis and wetting on solid surfaces: a unified description based on a virtual work principle

In order to account for diffusio-osmosis, Derjaguin proposed long ago that there is an excess pressure confined within a layer of typically a few nanometers in the vicinity of a solid surface immersed in a liquid and resulting from the interaction between the liquid and the surface. In the presence of a composition gradient in the liquid a confined pressure gradient parallel to the surface is therefore responsible for the diffusio-osmotic flow. This picture appears in contradiction with the contact theorem of colloidal science according to which such excess pressure does not exist.

We propose a theoretical description for calculating hydrodynamic flows in inhomogeneous liquids in the vicinity of solid interfaces which is consistent with the contact theorem. This approach is based on a Gibbs free energy and a virtual work principle for calculating the driving forces in the liquid due to inhomogeneous composition along a capillary and to the interaction with the solid interfaces. Our approach allows to show that the physics at play is the same in wetting or in diffusio-osmosis experiments, as one can go continuously from the latter to the former by making composition gradients sharper. We obtain an explicit expression for the diffusio-osmotic mobility which depends on the Gibbs free energy density in the vicinity of the interface and its dependance on the solute concentration in the liquid beyond the interfacial region, and which is inversely proportionnal to the liquid viscosity.

I. INTRODUCTION

The most simple description of hydrodynamic motion is provided by the Stokes equation which is valid when inertial effects are negligible, in the so-called low Reynolds number limit [START_REF] Landau | Fluid Mechanics[END_REF]. Even if this is the most simple limit of the Navier Stokes equation, this equation covers a broad range of phenomena and embodies a wealth of various physical situations. It is the relevant equation for polymer dynamics for describing Rouse relaxation or Zimm relaxation which takes into account long-ranges hydrodynamic interactions [START_REF] Doi | The Theory of Polymer Dynamics[END_REF]. Stokes equations are used also for describing the motion of electrically charged particles under the application of an external electric field [START_REF] Anderson | Colloid transport by interfacial forces[END_REF]. This motion is called electrophoresis. The same equations allow for describing the hydrodynamic flow created in a capillary when an electric field is applied [START_REF] Anderson | Colloid transport by interfacial forces[END_REF]. The electric field exerts a force in the vicinity of the capillary, where the liquid is electrically charged within the so-called Debye layer. This force pulls the liquids in the vicinity of the capillary and a so-called plug flow is created: velocity gradients are non zero within a few Debye length and beyond the hydrodynamic flow is uniform. If the capillary surface is uniformly charged and if the liquid is also homogeneous, no pressure is created by the applied electric field. When the surface is non-uniformly charged, a pressure gradient appears on larger scale and a so-called Poiseuille flow superimposes to the plug flow [4]. Stokes equations are relevant also for describing wetting phenomena [START_REF] De Gennes | Wetting: statics and dynamics[END_REF][START_REF] De Gennes | Capillarity and Wetting Phenomena. Drops, Bubbles, Pearls, Waves[END_REF]. A central issue of wetting is describing how a droplet spreads on a surface in order to reduce the interfacial energy which is the driving force of the effect. The reverse process, dewetting, is how a liquid recedes when it has been forcefully spread on a surface. These processes are described by Stokes equations in the presence of pulling forces resulting from surface tension imbalances at the triple line between the substrate, the liquid and the atmosphere or between the substrate and the two liquids when one considers a droplet imbedded in another liquid for instance.

Wetting or dewetting are a particular example of so-called "force-free" motions [START_REF] Stone | Engineering flows in small devices: Microfluidics toward a lab-on-a-chip[END_REF]. This name is given to motions which take place without the presence of an external applied field, contrary to the case of electrophoresis or electroosmosis where the driving force is an externally applied electric field. In the case of wetting or dewetting, the evolution and the motion are driven by internal forces. Other examples of "force-free" motions include diffusiophoresis where a solid particle moves in a liquid which has a gradient of composition, or thermophoresis where a solid particle moves in a liquid with a temperature gradient [START_REF] Anderson | Colloid transport by interfacial forces[END_REF].

Other kinds of so-called force-free motions may include spinodal decomposition in blends, in particular in polymer blends where phase separation processes result in motions without the presence of an applied field [START_REF] Clarke | Effect of shear flow on polymer blends[END_REF][START_REF] Cahn | Free energy of a non-uniform system. ı. interfacial free energy[END_REF].

Frequently, each of these phenomena are described in their particular way. Spinodal decomposition is described by the Cahn-Hilliard equation and its subsequent extensions: a gradient of chemical potential results in diffusion and possibly also in convection. The wetting dynamics of a droplet is described as a result of the imbalance of capillary forces at the triple line, which induces a Poiseuille-like Stokes flow in the droplet. Diffusiophoresis is described by the Derjaguin model [START_REF] Anderson | Colloid transport by interfacial forces[END_REF][START_REF] Anderson | Diffusiophoresis caused by gradients of strongly adsorbing solutes[END_REF][START_REF] Derjaguin | Kinetic phenomena in boundary films of liquids[END_REF][START_REF] Churaev | Surface Forces, chapter 11[END_REF][START_REF] Ajdari | Giant amplification of interfacially driven transport by hydrodynamic slip: Diffusio-osmosis and beyond[END_REF][START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF]. These authors assume that interfacial forces between the solid surface and the liquid, which have a range of up to a few tens of nanometers, create a local pressure in the vicinity of the interface. A gradient of composition along a capillary results in a gradient of pressure due to the interaction between the liquid and the solid interface. This gradient of pressure in the vicinity of the solid interface creates a diffusio-osmotic flow. This flow is a plug flow if the composition gradient in the vicinity of the interface is constant along the capillary. Calculating the pressure gradient requires a detailed description of forces on a local scale between the solute and the interface, including ionized charges in its vicinity when the interface becomes ionized. This description is complex and cannot be checked directly experimentally, except by measuring the diffusio-osmotic mobility which is the very effect that one aims at calculating. In any case, the Derjaguin picture used for describing diffusio-osmosis appears in contradiction with the well established contact theorem of colloidal science [START_REF] Grahame | The electrical double layerand the theory of electrocapillarity[END_REF][START_REF] Parsegian | On the electrostatic interaction across a sakt solution between two bodies bearing unequal charges[END_REF][START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF][START_REF] Ben-Yaakov | Beyond standard poissonboltzmann theory: ion-specific interactions in aqueous solutions[END_REF][START_REF] Adar | Osmotic pressure between arbitrarily charged planar surfaces: A revisited approach[END_REF], a consequence of which is that the pressure between two plates immersed in a liquid is uniform between the two plates.

The pressure becomes equal to the atmospheric pressure (if we assume that the system is equilibrated under the atmospheric pressure) when the distance between the two plates is larger than a few tens of nanometers, depending on the range of the interactions. In particular, the interfacial interactions between a single plate and the liquid does not result in the presence of an excess pressure in the vicinity of the solid surface. The pressure in that case is uniform and equal to the pressure under which the system is at thermodynamic equilibrium.

There is thus a need to reconcile what is known in colloidal science as regards to the equilibrium state in the vicinity of colloidal surfaces, the interaction between colloids on the one hand, and the description of the physics of diffusio-osmosis. The description of the latter, in its present state, is in contradiction with the former. We solve this problem by introducing a mesoscopic description of Stokes flow in inhomogeneous solutions and in the presence of solid interfaces as a relaxation process towards thermodynamic equilibrium in a way that is consistent with the Onsager general formulation. This description is based on a variational principal which relates the Gibbs free energy of the considered system to the driving forces in Stokes equation. This variational principle was introduced by Doi and Onuki [START_REF] Doi | Dynamic coupling between stress and composition in polymer solutions and blends[END_REF]. This variational principle is directly linked to the fact that the Stokes equations are Onsager-like equations [START_REF] Landau | Fluid Mechanics[END_REF][START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF] which drive the evolution of the system in out-of-equilibrium conditions and which relate linearly the fluxes to the thermodynamic forces. This description allows for describing convection in inhomogeneous liquids and flow in the presence of solid interfaces in situations corresponding either to wetting or dewetting or diffusio-osmosis. In particular there is no need for a confined pressure field in the vicinity of solid surfaces for creating diffusio-osmosis flows. Our description allows for a common formalism regarding wetting or diffusio-osmosis as relaxation processes towards thermodynamic equilibrium. The physics at play is the same in both types of experiments, as one can go continuously from the latter to the former by making the composition gradient sharper.

II. DERJAGUIN'S TREATMENT OF DIFFUSIO-OSMOSIS

Derjaguin considered the case of a solution bounded by a solid surface. The volume fraction of the solute is denoted by ψ. In the vicinity of the surface, there is an interaction potential Φ which decays with the distance z from the surface. This interaction potential exerts a body force due to the solute concentration which is given by f ext = -ψ∇Φ which we assume to be oriented in the direction opposite to the normal to the surface. The Stokes equations read [START_REF] Landau | Fluid Mechanics[END_REF]:

-∇p + ∇.η ∇v + ∇v + -ψ∇Φ = 0 ∇.v = 0 ( 1
)
where η is the viscosity, p is the pressure and v is the velocity field. The flow velocity is zero in the z-direction and is parallel to the interface. Let us suppose that the gradient of solute volume fraction is in the x-direction, and a decreasing function of x for being specific. By considering Eq.1 in the z-direction, one obtains then

-∂p ∂z -ψ ∂Φ(z) ∂z = 0
By integrating between the current z-position and far from the solid surface, one obtains an expression for the pressure term :

p(x, z) = -ψ(x)Φ(z) + p 0 ( 2
)
where p 0 is the atmospheric pressure under which the system is equilibrated (in the absence of solute concentration gradient). This pressure field is confined within a layer corresponding to the range of the potential interaction, which may be a few nanometers and up to a few tens of nanometers, depending on the nature of the interactions and the Debye layer of the solution as regard to ionic interactions. We see that this pressure field is present even in the absence of a gradient of solute concentration, provided the solute concentration is different from zero. If one considers a high volume fraction of solute, e.g. 1%, and if the interaction potential is 0.1 eV per molecular volume at the surface, the pressure is of order 10 7 Pa and decays to the atmospheric pressure under which the system is equilibrated. If the interface we consider is that of a capillary with diameter a few micrometers or more which connects two reservoirs, one with a high solute concentration and the other one without solute, a concentration gradient establishes along the capillary in the vicinity of its surface. The surface of the latter can be considered as flat if the diameter is much larger than the range of the potential Φ which we assume. Then, the pressure gradient is described by Eq.2 and

Eq.1 becomes:

∂ψ(x) ∂x Φ(z) + η ∂ 2 v x ∂z 2 = 0 (3)
Let us assume that Φ is given by

Φ(z) = Φ(0) exp( -z λ )
This equation allows to calculate the flow velocity profile

v x (z) = λ 2 η ∂ψ(x) ∂x Φ(0) exp( -z λ ) -1 (4)
electro-osmosis for instance [4]. This is a standard behavior when forces act on the liquid only within a thin layer in the vicinity of a solid surface, as it is a case for the considered diffusio-osmotic flow. The driving force in Derjaguin's picture is the pressure field in the vicinity of the capillary wall.

However, the term p(x, y) is difficult to interpret. Even in the absence of a concentration gradient, it corresponds to a local pressure that can be of typical order of magnitude 10 6 -10 7

Pa. This pressure should be maximal at the very interface between the solid and the liquid and decay towards the atmospheric pressure at a distance corresponding to the range of the interaction potential Φ(z). As mentioned above, this picture is in contradiction with the current understanding regarding interactions between colloidal particles in solutions, or regarding the thermodynamical state of a liquid in the vicinity of a colloidal particle at thermodynamic equilibrium. For instance, if one considers two parallel plates in an electrolyte solution [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF], the Poisson-Boltzmann model predicts that the repulsive pressure between the two plates is homogeneous in the volume bounded by these two plates and is given by

P (z, D) = k B T ρ 0 (D) + p 0
where k B is the Boltzmann constant, T is the temperature, D is the distance between the two plates, ρ 0 (D) the counterion density mid-way between the plates and z the coordinate in the direction normal to the plates. In particular the pressure is uniform and decays to zero when the plates are separated by a distance larger than the range of the molecular interactions, which are typically of order a few nanometers up to a few tens of nanometers.

In a capillary for which the diameter is larger than a micrometer, this pressure is uniform and equal to the atmospheric pressure p 0 . This result is known as the so-called contact theorem [START_REF] Grahame | The electrical double layerand the theory of electrocapillarity[END_REF][START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF][START_REF] Ben-Yaakov | Beyond standard poissonboltzmann theory: ion-specific interactions in aqueous solutions[END_REF] and is exact, beyond the Poisson-Boltzmann approach and regardless of the nature of the interactions.

The presence of a layer with a pressure larger than the atmospheric pressure under which the system is equilibrated raises indeed some questions. The first one is the meaning of the grand-canonical equilibrium under an imposed pressure (e.g. atmospheric pressure p 0 ) if some regions of the considered system have a higher pressure than the nominal pressure p 0 as a result of the interactions between parts of the systems. For instance, the thin layer with a higher pressure in Derjaguin's picture should relax by slightly expanding to decrease the pressure towards p 0 , thereby reducing the Gibbs free energy of the system. The very presence of this thin layer with a higher pressure seems to contradict the very concept of thermodynamic equilibrium under an imposed pressure.

Our point of view is that Derjaguin's calculation takes into account only a part of the stress tensor, corresponding to molecular interaction between the long-ranged interaction between the solid surface and the solute. However, this is not correct. When one considers the thermodynamics of dielectric media one does not consider separately the stress associated to intermolecular interactions and the stress of the electric field itself. One considers only the whole. This is the meaning of the equilibrium condition discussed in Landau and Lifshitz [START_REF] Landau | Electrodynamics of Continuous Media[END_REF],

for which the field and the density of the sample relax together to build the thermodynamic equilibrium. It is explicitly stated that there is no body force in the dielectric. On the contrary, in Derjaguin's picture, the interaction between the solid surface and the liquid are treated as external body forces acting on the liquid. In a dielectric, all forces on a elementary volume are transmitted through the boundaries by the local stress tensor σ which satisfies thus the condition [START_REF] Landau | Electrodynamics of Continuous Media[END_REF] :

∇.σ = 0 (5)
Derjaguin's force violates this condition by introducing a body force. The thermodynamic equilibrium is described by minimizing the contribution of all the interactions, that of the field and the intermolecular forces that cannot be considered separately. This is what makes necessary the introduction of the polarizability of a material. This is the case for instance when one considers the electro-striction effect in a dielectric. The body force of the Derjaguin's treatment of the diffusio-osmosis is only a part of the interaction. One cannot consider that only a part of the local stress tensor acts on the liquid without taking into account all the contributions to the stress tensor. And when one take all the contributions, the pressure is that under which the system is equilibrated, that is the atmospheric pressure p 0 . As a consequence, the Derjaguin's picture breaks down for describing diffusio-osmosis.

One needs to propose another approach for explaining the diffusio-osmotic effect consistently with what is known regarding the equilibrium thermodynamics of liquids in the vicinity of colloids.

III. HYDRODYNAMICS

We introduce in this section a formalism for Stokes equations in out-of-equibrium liquids, such as inhomogeneous ones, based on a virtual work principle. It allows to obtain equations which describe the Stokes flow as a relaxation process towards thermodynamic equilibrium.

In the following we use a non-specified Gibbs free energy. It is made dimensionless by considering the Gibbs free energy per monomer, by dividing the physical Gibbs free energy by the thermal energy, and by setting as unit length scale the monomer length or the molecular size in simple liquids. In an out-of-equilibrium situation, one can calculate the stress as a driving force in a liquid according to a principle of virtual work. The total Gibbs free energy is given by

G = g(ψ(r), ∇ψ(r))d 3 r
g is the density of Gibbs free energy. It is a function of ψ and ∇ψ. The dependance of g on ∇ψ allows for taking into account the finite range of the interactions. Following Doi and Onuki [START_REF] Doi | Dynamic coupling between stress and composition in polymer solutions and blends[END_REF], let us consider a virtual displacement in the liquid so that r is changed into r + δr. The quantities ψ and ∇ψ are varied by an amount δψ = -δr.∇ψ;

δ∇ψ = ∇δψ

The density of Gibbs free energy may be assumed to be the sum of two contributions :

g(ψ, ∇ψ) = g (0) (ψ) + g (1) (∇ψ)
Then the variation of Gibbs free energy is given by :

δG = δg(r)d 3 r = dg (0) dψ δψ + dg (1) d∇ψ .δ∇ψ d 3 r
By substituting Eq.6 in the previous equation, and by integrating by parts the second term, one obtains:

δG = -δr.∇ψ dg (0) dψ -∇. dg (1) d∇ψ d 3 r (7)
We assume that the term dg (1) d∇ψ δr.∇ψ is zero on the boundaries. This is the case if the flow we consider is zero at the boundaries, or if the fluid is homogeneous at these boundaries. If this is not the case, an additional contribution at the boundaries should be added. The stress is related to the variation of Gibbs free energy by :

δG = -δr.∇.σd 3 r (8)
One obtains thus an expression for the hydrodynamic stress:

∇.σ = dg (0) dψ -∇.
dg (1) d∇ψ ∇ψ = µ∇ψ

One may introduce the quantities

µ (0) = dg (0) dψ µ (1) = dg (1) dψ µ (0)
is the usual chemical potential. µ (1) is a functional derivative which contains the contribution in ∇ψ to the Gibbs free energy. This is this term which gives rise to surface tension in inhomogeneous liquids. The force term resulting from thermodynamic forces has to be included into the Stokes equation for describing hydrodynamic flow in out of equilibrium conditions. An explicit expression for µ is given in Appendix A for a specific Gibbs free energy.

The stress tensor introduced above, which we denote here σ G , allows for calculating the free energy release associated with a flow field v. The Stokes equation allows for calculating the dissipation due to the flow. When a liquid is submitted to a body force f ext , the dissipated power P reads :

P = -d 3 r∇.σ Hydr .v = f ext .v (9)
When the forces are internal, the dissipation is equal to the opposite of the rate of relase of free energy σ G , and the equations read

P = - dG dt = v.∇σ G d 3 r (10)
One obtains then

∇.σ Hydr + ∇.σ G = 0
This equation ensures that the relaxation towards equilibrium is consistent with Onsager theory. The fluxes (∇v) are proportional to the thermodynamic forces -σ G . The free energy release rate is equal to the dissipation rate and this equality is local. The basic equation regarding the hydrodynamic flow is therefore:

-∇p + ∇.η ∇v + ∇v + + f ext + µ∇ψ = 0 ∇.v = 0 ( 11
)
where we added a possible external force f ext which would drive the system out-of-equilibrium.

The term µ∇ψ corresponds to internal forces.

IV. HYDRODYNAMICS AS A BRANCH OF OUT-OF-EQUILIBRIUM STATIS-TICAL PHYSICS

Navier-Stokes equations are described in e.g. Landau and Lifshitz [START_REF] Landau | Fluid Mechanics[END_REF] as a branch of out-of-equilibrium statistical physics. Out-of-equilibrium systems have been described by Onsager by introducing an out-of-equilibrium entropy. This entropy is smaller than that at equilibrium. This entropy is expressed as a function of coarse-grained variables (x a ). The derivative of the entropy according to each of these variables provides the thermodynamic forces X a [START_REF] Landau | Fluid Mechanics[END_REF][START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF][START_REF] Landau | Statistical Physics, part 2[END_REF] :

X a = - ∂S ∂x a (12)
Then, the variables x a relax towards equilibrium according to Onsager equations:

dx a dt = -Σ b γ a,b X b (13)
The matrix (γ a,b ) is symmetric positive definite. This equation describes how the entropy increases in order to reach the thermodynamic equilibrium at which it is maximum. The Navier-Stokes equation can be considered within the Onsager formalism. One needs to introduce an explicit spatial aspect to these equations. The entropy production reads then

dS dt = -Σ a X a ẋa d 3 r (14)
From the expression of the entropy production derived from the Navier-Stokes equations, that are the conservation of momentum, the conservation of energy, the conservation of mass and of constituents, it is then possible to identify the thermodynamic forces X a and the fluxes ẋa [START_REF] Landau | Fluid Mechanics[END_REF]. The thermodynamic forces as derived from the Navier-Stokes equations appear as equal to the thermal gradient and to the chemical potential gradient.

As a consequence, the Navier-Stokes equations appear as the relaxation process for newtonian liquids when they are out of thermodynamcial equilibrium and that their entropy is not at the maximum possible value. The relaxation processes as described in Landau and

Lifshitz [START_REF] Landau | Fluid Mechanics[END_REF] or in De Groot and Mazur [START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF] are thermal and solute diffusion. No coupling between the out-of-equilibrium thermodynamical state and convection has been introduced.

As compared to these standard treatments, we make two changes: a) First of all, instead of considering an out-of-equilibrium entropy S, we deal with an out-of-equilibrium Gibbs free energy for which similar approaches hold and are most frequently discussed in the context of the linear response theory [START_REF] Pottier | Nonequilibrium Statistical Physics[END_REF][START_REF] Reichl | A Modern Course in Statistical Physics[END_REF]. The reasons are the following. The general treatments by introducing an out-of-equilibrium entropy considers that small volume elements of the liquids can be considered as closed systems. Then, their internal state evolves by maximising their entropy. The Navier-Stokes equations allow for describing on longer time scales their exchange of energy and constituents with their neighbours by introducing energy and constituents laws conservation laws. When considering wetting or diffusio-osmosis problems, the full Navier-Stokes equations are not necessarily required. The small volume elements can be considered as thermostated. In that case, the relevant out-of-equilibrium function is the local excess of free energy. In addition, if the pressure can be considered as imposed, the relevant function is the local excess of Gibbs free energy. Instead of deriving the relaxation processes thanks to an entropy production, the relaxation processes are described by the release of an excess of Gibbs free energy [START_REF] Pottier | Nonequilibrium Statistical Physics[END_REF][START_REF] Reichl | A Modern Course in Statistical Physics[END_REF].

This is what is done for instance for considering relaxation processes in polymer rheology where the corresponding function is called the dynamical free energy by Doi and Edwards [START_REF] Doi | The Theory of Polymer Dynamics[END_REF] and is essentially an excess of Gibbs free energy. b) We do not consider solute diffusion, because it is either very fast in the direction normal to the considered solid surfaces, or too slow in the direction parallel to the solid surfaces that we consider. We do not consider either thermal diffusion because energy is not conserved in the canonical ensemble and is not conserved either in the Stokes equation.

Energy is exchanged rapidly with the thermostat.

However, extension of the present theory for describing thermal diffusion and solute diffusion, following the lines presented in Landau and Lifshitz [START_REF] Landau | Fluid Mechanics[END_REF] is possible. Note that it is what has been done by Julicher and Prost [START_REF] Julicher | Generic theory of colloidal transport[END_REF] without discussing the effect of solute on convection or the effect of interfaces on the flow and how surface tension gradient come into play.

The entropy production function as calculated thanks to the Navier Stokes equation [START_REF] Landau | Fluid Mechanics[END_REF][START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF] is of importance because it relates the relevant fluxes to the relevant thermodynamic forces and allow them to be to identified in a way that is consistent with the general formalism of Onsager theory. We do the same as regards to convection, which has not been considered in this way to the best of our knowledge.

Starting from Eq.10 which has the general required form, we deduce that v are the fluxes and that ∇σ G = µ∇ψ is the thermodynamic force. The general solution is

v = M∇.σ G ( 15
)
where M is a mobility tensor. The only solution for the flow which has the required symmetry, that is which is invariant by translation and rotation is that given by the Stokes equation [START_REF] Landau | Fluid Mechanics[END_REF]. Hence the flow is given by the solution of the Stokes equation 11.

V. HYDRODYNAMICS WITHOUT BOUNDARY CONDITIONS

Let us consider the Stokes equations in the presence of a chemical composition gradient in the unbounded 3D space. In the absence of short-ranged interactions, i.e. when there is no surface tension term, or gradient term in the free energy, a heterogeneous composition does not create any flow in an unbounded fluid. Indeed, in these conditions, the quantity µ is a function of ψ only and not on ∇ψ. The corresponding driving force in Eq. 11 due to composition gradient can be written as

µ(ψ)∇ψ = ∇g (0) (ψ)
since we have

dg (0) (ψ) dψ = µ(ψ)
The Stokes equation reads then

-∇ p -g (0) (ψ) + ∇.η ∇v + ∇v + = 0 (16) ∇.v = 0
From these two equations, one obtains the following equation:

-∇ p -g (0) (ψ) + η∆v = 0 ∆ p -g (0) (ψ) = 0
where ∆ is the laplacian operator. Given the boundary conditions of uniform pressure and solute concentration at infinity and of no flow at infinity, one obtains the following solution:

p -g (0) (ψ) = p 0 ; v = 0
The physical pressure is the entire term p -g (0) (ψ) and is uniform. The quantity p is not the total pressure, it is just the Lagrange multiplier which ensures incompressibility of the flow. The term µ∇ψ which appears as a body force in the Stokes equation would result in a longitudinal flow with non zero divergence. This effect must be compensated to maintain the purely transverse nature of the flow. Without non-local interactions embodied with the ∇ψ dependence of the Gibbs free energy, non-homogeneous composition cannot create convection in an unbounded fluid. The relaxation towards equilibrium occurs entirely by diffusion. For instance, if we consider a salt gradient concentration in water, if there is no long range interaction and that the chemical potential depends only on the local concentration, such a gradient cannot create flow. In fact the local Lagrange multiplier p adjusts so that p -g (0) (ψ) is uniform and no flow takes place. This issue has been discussed with different arguments by Julicher and Prost [START_REF] Julicher | Generic theory of colloidal transport[END_REF].

On the other hand, the term g (1) contributes to convection flow in the presence of concentration gradients. The corresponding term is responsible for the surface tension between two different A-rich and B-rich domains [START_REF] Cahn | Free energy of a non-uniform system. ı. interfacial free energy[END_REF][START_REF] Safran | Statistical Thermodynamics of Surfaces, Interfaces, and Membranes[END_REF]. Thus, all the convection in such out-ofequilibrium systems is due to surface tension-like effects. All the contributions to convection come from the dependence on ∇ψ of the Gibbs free energy 1 .

Finally, another way to see that g (0) contributions cannot give rise to convection is the following. Under an infinitesimal displacement, the new Gibbs free energy is given by :

G = g(r)d 3 r = g (0) (ψ(r -δr))d 3 r
One can consider the change of variable r = r-δr. Since the allowed displacements preserve the volume the corresponding Jacobian of the change of variable is 1. One has thus:

G = g (0) (ψ(r -δr))d 3 r = g (0) (ψ(r))d 3 r
Virtual displacements which preserve the volume cannot change the Gibbs free energy when the latter depends only on local variables. As a consequence, there can be no convection due to purely local contribution to the Gibbs free energy such as those corresponding to ideal solutions. One needs to introduce short-ranged interaction through concentration gradient contributions.

As a consequence, when there is no short-ranged interactions, the concentration evolves towards equilibrium only by diffusion or by an imposed convection, but composition gradients do not cause convection. In general, the complete evolution equation for the concentration is a convection-diffusion equation. 1 It can be checked in Fourier space that the term µ (0) gives contributions to the Stokes equation which are purely longitudinal which are cancelled by the pressure term p to ensure incompressibility. On the other hand, a µ (1) term like ∆ψ gives rise to transverse contributions to the Stokes equation, thereby giving rise to convection.

Having established the hydrodynamic equation in an unbounded fluid, we need to establish the equations in the presence of boundaries. Theses boundaries may be the wall of a capillary, or the surface of a particle. The boundary conditions must take into account two important physical effects. The first one is the interaction between the solid surface and the liquid. It is important to stress that these interactions have a finite range. Though small, this range is not zero and typically of order a few nanometers [START_REF] Anderson | Colloid transport by interfacial forces[END_REF]. This is of fundamental importance as regard to wetting dynamics for instance. The physical origin of these interactions are van der Waals interactions, or ionic interactions for instance. It is not possible to describe properly wetting dynamics without taking into account this non-zero range of the interactions. The second feature that is of major importance for describing the flow of the liquid in the vicinity of a solid surface is the slipping length. The macroscopic description assumes a no-slip boundary condition for the flow. However, again, if one wants to describe and understand wetting dynamics, one needs to assume that there is a non-zero slipping length. This slipping length in the case of simple liquids as regard to wetting dynamics experiments may be assumed to be of order one nanometer [START_REF] De Gennes | Wetting: statics and dynamics[END_REF]. A larger slipping length has been discussed in the literature, up to a few tens of nanometer [START_REF] Ajdari | Giant amplification of interfacially driven transport by hydrodynamic slip: Diffusio-osmosis and beyond[END_REF] but the issue is still under debate [START_REF] Lauga | Handbook of Experimental Fluid Dynamics[END_REF].

For describing the effect of the interface on the hydrodynamic flow, we introduce an additional contribution to the Gibbs free energy density in the vicinity of an interface. We denote this contribution by Γ(ψ, r), where ψ is the solute volume fraction in the liquid slightly beyond the interfacial region. The range of Γ is typically of few nanometers and up to a few tens of nanometers depending on the thickness of the interfacial layer. This is the contribution of the solid surface to the free energy of the liquid in its vicinity. The contribution of the interface to the Gibbs free energy reads

G Int = Γ(ψ, r)d 3 r
The quantity Γ(ψ, r) must be understood in the following way: we assume that at some distance from the interface, the solute volume fraction ψ is imposed by the flow. Then, the interfacial layer equilibrates locally to assume a certain profile corresponding to the local minimum of Gibbs free energy (such as that discussed in Appendix) given the specific nature of the interactions between the solute, the liquid and the solid surface and the imposed conditions ψ of solute concentration at a few correlation lengths from the solid surface [START_REF] Safran | Statistical Thermodynamics of Surfaces, Interfaces, and Membranes[END_REF].

This decription relies on the following assumptions: the local equilibration is fast enough as compared to the flow; the concentration gradient imposed by the flow and parallel to the solid surface corresponds to a length scale much larger than the thickness of the interfacial region between the solid and the liquid; the bulk of the capillary acts as a reservoir for equilibrating the interfacial layer. The local equilibration is assured by a diffusion process which we assume thus to be faster on the scale of the interfacial layer than the disturbance caused by the flow.

Considering the same principle of virtual work in the vicinity of the boundary leads to the Stokes equation in the vicinity of a solid interface. Let us consider a virtual displacement in the liquid so that r is changed into r + δr. The quantity ψ is varied by an amount δψ = -δr.∇ψ(r);

Then

δG Int = -δr.∇ψ(r) ∂Γ(ψ, r) ∂ψ d 3 r (17)
The derivative of Γ is calculated with respect to the imposed solute concentration beyond the interfacial layer. By adding the corresponding term in the Stokes equation, one obtains the Stokes equation in the presence of an interface:

-∇p(r) + ∇.η ∇v(r)

+ ∇v + (r) + µ(r)∇ψ(r) + ∂Γ(ψ, r) ∂ψ ∇ψ(r) = 0 (18)
The total chemical potential is given by

µ tot = µ + ∂Γ(ψ, r) ∂ψ (19)
The term µ contains all the contribution for the bulk, the second one contains the contribution which results from the interaction with the solid interface. The second term dΓ(ψ) dψ is different from zero only in the vicinity of the interface. The variation of surface tension ∆γ(ψ) between the solid surface and the liquid with solute concentration ψ and the liquid with solute concentration zero is given by ∆γ

= ∞ 0 (Γ(ψ, z) -Γ(0, z)) dz One obtains then dγ dψ = ∞ 0 ∂Γ(ψ, z) ∂ψ dz = ∞ 0 Ψ(ψ, z)dz
where γ is the surface tension between the liquid and the solid. We have introduced

Ψ(ψ, z) = ∂Γ(ψ, z) ∂ψ A.

Diffusio-osmosis without slipping length

Let us consider a capillary of sufficiently large diameter so that its surface can be considered locally flat. Let us suppose that the liquid has a uniform gradient of composition which induces a uniform gradient of surface tension between the liquid and the capillary along the x-direction. The normal to the flat surface is the local z-direction. Let us suppose that the bulk chemical potential µ in Eq.18 does not contain µ (1) -like contributions. µ (0) -like contributions cannot give rise to convection and can be discarded from the convection equations.

The flow equation 18 reduces thus to

-∂p ∂x + η ∂ 2 v x ∂z 2 + ∂ψ ∂x ∂Γ ∂ψ = 0 ( 20
)
Let us assume that the gradient of concentration ψ is constant along the capillary as well as Ψ which is the contribution of the interfacial interactions to the local chemical potential.

By taking the derivative of Eq.20 with respect to x and by using the incompressibility condition and the invariance by translation of the other terms, we deduce that the pressure p satisfies the following equation :

∂ 2 p ∂x 2 = 0
If the pressures at both extremities of the capillary are identical, e.g. the atmospheric pressure, we deduce from this equation that the pressure is uniform within the capillary and equal to the atmospheric pressure.

This equation allows to calculate the flow velocity profile

∂v x ∂z (z) = -1 η ∂ψ ∂x z 0 Ψ(z )dz - ∞ 0 Ψ(z )dz = 1 η ∂ψ ∂x ∞ z Ψ(z )dz (21)
We observe thus that the shear rate is a decreasing function of the distance to the interface and is zero at distances larger than the range of the thermodynamic perturbation of the liquid by the interface. One obtains

∂v x ∂z (0) = 1 η ∂ψ ∂x ∞ 0 Ψ(z )dz
If we denote by a the range of Ψ, and assuming a no slip boundary condition at z = 0, one obtains that the velocity of the flow at a distance z > a can be approximated by

v x (z) ≈ a η ∂ψ ∂x ∞ 0 Ψ(z )dz
The velocity field obtained by solving Eq.21 and assuming a no slip boundary condition at z = 0 is given by:

v x (z) = 1 η ∂ψ ∂x z 0 ∞ z Ψ(z )dz dz (22)
Equation 22 (together with Equation 18) is the main result of this article. It represents a so-called plug-flow observed also in electroosmosis for instance. This is a standard behavior when forces act on the liquid only within a thin layer in the vicinity of a solid surface, as it is a case for the considered diffusio-osmosis flow. We see that the flow velocity depends on the range of the interfacial forces a.

The term ∂ψ ∂x ∂Γ ∂ψ in Eq. 20 represents a pulling force parallel to the interface. This term is analogous to the term ρE ∞ in electroosmosis [START_REF] Anderson | Colloid transport by interfacial forces[END_REF]4] where ρ is the density of charge in the vicinity of a solid surface which is non zero within the Debye layer, and E ∞ is the applied electric field in the direction parallel to the capillary. In the problem we consider here, the forces are internal and result from the interaction between the liquid and the solid. They appear as parallel to the surface: the surface pulls the liquid in order to reduce the Gibbs free energy of the liquid, taking into account the interaction between the solid and the liquid.

These forces are tangential and do not result in the appearance of a confined pressure, nor are they the result of a preexisting confined pressure in the vicinity of the liquid-solid interface.

As we discussed, the diffusio-osmotic flow is a so-called plug flow when the tangential force is constant. When this is not the case, one observes the superimposition of local plug flows and of a Poiseuille flow on larger scale. This situation has been discussed in the case of electroomsosis e.g. in reference [4]. Let us assume that Ψ(z) is given by

Ψ(z) = Ψ(0) exp( -z a )
One obtains then dγ dψ = aΨ(0)

These equations allow to calculate the flow velocity profile

v x (z) = a 2 η ∂ψ ∂x Ψ(0) 1 -exp( -z a )
The diffusio-osmotic velocity in the capillary is then given by

V = a 2 η ∂ψ ∂x Ψ(0) = a η ∂ψ ∂x dγ dψ
The diffusio-osmotic mobility defined by V = µ∇ψ is then given by

µ osm chem = a η dγ dψ
We obtain that the diffusio-osmotic mobility is proportional to the range of the interaction and to the derivative of the liquid-solid surface tension with respect to the solute concentration. When the gradient of solute concentration is zero, Eq.18 admits a no flow and uniform pressure solution (the atmospheric pressure p 0 ), contrarily to the description of diffusio-osmosis by the Derjaguin's model [START_REF] Anderson | Colloid transport by interfacial forces[END_REF][START_REF] Derjaguin | Kinetic phenomena in boundary films of liquids[END_REF][START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF][START_REF] Anderson | Motion of a particle generated by chemical gradients part 1. non-electrolytes[END_REF], as can be seen by reference to 2.

The fact that the diffusio-osmotic mobility is related to the gradient of surface tension has been recognized by many authors [START_REF] Anderson | Motion of a particle generated by chemical gradients part 1. non-electrolytes[END_REF][START_REF] Ruckenstein | Can phoretic motions be treated as interfacial tension gradient driven phenomena ?[END_REF]. However, the calculations by Ruckenstein or by

Anderson et al derive from the Derjaguin's picture and rely on a confined pressure gradient as a driving force. Levich describes also the flow as due to surface tensions gradient for a liquid-liquid interface corresponding to a Marangoni effect [START_REF] Levich | Physicochemical Hydrodynamics[END_REF] but this description cannot be extended to a solid-liquid interface which requires taking into account a description of the interfacial layer. Indeed, the effect of the structure and the perturbation of the density of Gibbs free energy on the hydrodynamic equation embodied in Equation 18 which leads to Equation 22is not present in Levich's description. More recently, Marbach et al [START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF] and Liu et al [START_REF] Liu | Pressure gradients failto predict diffusio-osmosis[END_REF] questions the role of the local pressure gradient and conclude that the osmotic flow is more probably related to the chemical potential gradient. Equations 18 and 22 provide an explicit dependence of the driving forces in the liquid and of the resulting diffusio-osmotic flow. This 3D driving force in the vicinity of the surface is, apart from a prefactor, the derivative of the local excess Gibbs energy density with respect to the imposed solute concentration beyond the interfacial layer. Our expression makes explicit the detailed nature of the interaction between the surface and the liquid which is responsible for the flow. Note that this is only the latter which is responsible for this effect since we have shown that local terms for the chemical potential such as ln c where c is the concentration, or any other term corresponding to a so-called ideal solution do not contribute to the flow.

It is important to note that the driving force that we obtain is µ∇ψ in Equation 11.

Araki and Tanaka [START_REF] Araki | Wetting-induced depletion interaction between particles in a phaseseparating liquid mixture[END_REF] introduced a similar Stokes equation but the driving force in their article is -ψ∇µ. As regards to our derivation of the driving force, Araki and Tanaka's equation may be obtained by an additional integration by part following Equation 7, using the fact that ∇.δr = 0 due to incompressibility. However, this integration by parts should come with contributions at the boundaries which are :

-ψ dg (0) dψ -∇. dg (1) d∇ψ + Ψ(ψ(x), z) δr (23)
The contribution of this term is zero on the boundary corresponding to the solid interface since δr is zero there. But the contribution of this term is non zero in general. This is the case in particular if we consider diffusio-osmosis along a capillary which connects two reservoirs

(1) and ( 2) of respective concentrations ψ 1 > ψ 2 . The quantities related to g (1) in Eq.23 may be considered as zero at the boundaries because the reservoirs have a homogeneous concentration, but the two quantities ψ dg (0) dψ corresponding to the two reservoirs are different. The integration by parts which leads to this different driving force (as compared to µ∇ψ) is unwarranted, except in the special case of periodic boundary conditions relevant to the application considered by Araki and Tanaka [START_REF] Araki | Wetting-induced depletion interaction between particles in a phaseseparating liquid mixture[END_REF]. In this case the difference between µ∇ψ and -ψ∇µ is unimportant since the contributions at the boundaries cancel. An analogous term ψ∇µ has been considered also by Marbach et al [START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF] and by Liu et al [START_REF] Liu | Pressure gradients failto predict diffusio-osmosis[END_REF], without specifying µ as regards to the interfacial interactions in the same way as we do though. If one uses this term as a driving force in the Stokes equation, one obtains a description very similar to that of Derjaguin since this term leads to the appearence of a confined pressure in the vicinity of the capillary, as shown in Appendix C. In principle this confined pressure should be cancelled by the effect of the boundary terms just mentioned which are absent though in all descriptions in the literature [START_REF] Anderson | Colloid transport by interfacial forces[END_REF][START_REF] Derjaguin | Kinetic phenomena in boundary films of liquids[END_REF][START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF][START_REF] Anderson | Motion of a particle generated by chemical gradients part 1. non-electrolytes[END_REF][START_REF] Liu | Pressure gradients failto predict diffusio-osmosis[END_REF][START_REF] Araki | Wetting-induced depletion interaction between particles in a phaseseparating liquid mixture[END_REF]. As a consequence, the term -ψ∇µ cannot be considered as the local driving force in the Stokes equation [START_REF] Araki | Wetting-induced depletion interaction between particles in a phaseseparating liquid mixture[END_REF]. We propose that the local driving force is µ∇ψ.

B. Diffusio-osmosis and slipping length

One can also describe these effects by taking into account a non-zero slipping length whereas the range of the interactions are zero, that is the effect of the interaction between the solid and the liquid are purely interfacial. The flow equation is given by Eq.20. We assume here that the interfacial perturbation due to the solid-liquid interface is non zero only at the interface and that the flow is characterized by a slipping length b [START_REF] De Gennes | Capillarity and Wetting Phenomena. Drops, Bubbles, Pearls, Waves[END_REF]. If the interface is located at z = 0, Γ as a 3D function may be written as Γ(ψ(r), z) = Γ(ψ(x, y, 0))δ(z) where δ(z) is the Dirac function.

Then, Eq.20 can be solved by describing a virtual velocity profile for z > -b whereas the solid-liquid interface is located at z = 0. The slipping length b is the length for which the flow, extrapolated in the negative z region cancels [START_REF] De Gennes | Capillarity and Wetting Phenomena. Drops, Bubbles, Pearls, Waves[END_REF]. The flow is then described with a no-slip boundary condition at z = -b and with pulling forces located at z = 0. One obtains then a plug like velocity profile where the velocity is uniform for z > 0 and is given by

v x (z) = b η ∂ψ ∂x dΓ dψ
The diffusio-osmotic mobility defined by V = µ∇c is given by

µ osm chem = b η dΓ dψ
We obtain here similar results as above where the range of the interaction a is replaced by a slipping length b. More details are given in Appendix B.

We see thus that the diffusio-osmotic mobility is proportional either to the slipping length or to the range of the interaction. To understand this effect, one needs to take into account that the tangential forces are exerted on the fluid at a different location where a no-slip boundary condition holds. The general result involves thus a combination of these two length scales as discussed e.g. in reference [START_REF] Ajdari | Giant amplification of interfacially driven transport by hydrodynamic slip: Diffusio-osmosis and beyond[END_REF], though the interpretation of the origin of the effect is different.

C. Wetting

Wetting can also be described by using this formalism. Let us consider Eq.20. Wetting of the pure liquid with the solid surface2 . Thus, when integrated over the small region of size a in the direction normal to the surface and of size ξ in the direction parallel to the surface (where ξ is the thickness of the liquid A-liquid B interface) where the pulling body force of Eq.18 is non zero, one obtains that the total pulling force due to the interaction with the solid substrate is equal to γ B -γ A . This is schematized in Figure 2. We recover thus the standard description of wetting when adding the contribution γ A,B cos θ due to the A-B surface tension provided by the third term of 18 where θ is the contact angle of the droplet with the solid substrate. This standard description may be obtained directly by a virtual work principle description of this problem in terms of interfacial free energy.

As a consequence, the formalism introduced here allows for describing wetting or dewetting dynamics as a special case of diffusio-osmosis and to obtain the standard description of wetting in the literature [START_REF] De Gennes | Wetting: statics and dynamics[END_REF][START_REF] De Gennes | Capillarity and Wetting Phenomena. Drops, Bubbles, Pearls, Waves[END_REF].

VII. DISCUSSION AND CONCLUSION

We have introduced a formalism which allows to describe diffusio-osmosis in a way which is consistent with the contact value theorem of colloidal science [START_REF] Grahame | The electrical double layerand the theory of electrocapillarity[END_REF][START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF][START_REF] Ben-Yaakov | Beyond standard poissonboltzmann theory: ion-specific interactions in aqueous solutions[END_REF][START_REF] Adar | Osmotic pressure between arbitrarily charged planar surfaces: A revisited approach[END_REF]. According to this theorem, there is no excess pressure in the vicinity of a flat surface immersed in a liquid which would result from the interaction between the surface and the liquid, whereas it was assumed by Derjaguin that such a pressure is the cause for diffusio-osmostic flows. More recently, the relevance of the assumed pressure gradient as the cause of the diffusio-osmotic flow has been questionned by Marbach et al [START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF] and by Liu et al [START_REF] Liu | Pressure gradients failto predict diffusio-osmosis[END_REF]. These authors conclude that the chemical potential gradient is probably a more relevant quantity. Our formalism provides an explicit expression for the driving force of the diffusio-osmotic flow confirming that this flow is not related to a local pressure gradient which, we argue in this manuscript,

does not exist at all. Note that a pressure gradient arises on larger scale when the driving forces are not invariant by translation along the capillary. In this case Poiseuille-like flows superimpose to the plug-like flow in an analogous way to what has been described in the case of inhomogeneous electroosmosis [4].

Our formalism is based on a principle of virtual work which makes apparent that the driving force is the consequence of an excess of Gibbs free energy release. This excess of Gibbs free energy as compared to an equilibrium situation is located in the vicinity of the solid-liquid interface. In the presence of a gradient of composition of the liquid along the capillary it gives rise to tangential body forces within the interfacial layer. In a coarsegrained picture, this excess of free energy may be related to a gradient of surface tension between the liquid and the solid. What is key in this picture is that the driving forces are purely transverse and have a spatial extension. They are not applied exactly where a no-slip boundary condition holds. The mobility is proportional either to the range of the interaction or to a slipping length and may be a combination of both in the general case. This formalism makes more apparent the physical nature of the driving force by which the excess of Gibbs free energy is released in a way which is consistent with the Onsager formalism.

This formalism can also be applied for describing wetting and/or dewetting which appears as a limiting case of diffusio-osmotic flows with a sharp composition gradient between two different liquids. In wetting situations, our approach is consistent with the driving forces being the surface tensions between both liquids and between each of the liquids and the solid substrate. No reference is made to a possible high pressure confined region at the liquid/solid interfaces. Our point of view is that the standard picture for describing wetting is correct and that the Derjaguin's picture is not correct at this regard. The variationnal principle introduced by Doi and Onuki [START_REF] Doi | Dynamic coupling between stress and composition in polymer solutions and blends[END_REF] and which we use in this context allows to propose a unified and consistent picture for decribing wetting [START_REF] De Gennes | Wetting: statics and dynamics[END_REF][START_REF] De Gennes | Capillarity and Wetting Phenomena. Drops, Bubbles, Pearls, Waves[END_REF] and diffusio-osmosis [START_REF] Anderson | Colloid transport by interfacial forces[END_REF][START_REF] Derjaguin | Kinetic phenomena in boundary films of liquids[END_REF][START_REF] Churaev | Surface Forces, chapter 11[END_REF][START_REF] Anderson | Motion of a particle generated by chemical gradients part 1. non-electrolytes[END_REF] consistently with some key results of colloidal physics [START_REF] Grahame | The electrical double layerand the theory of electrocapillarity[END_REF][START_REF] Parsegian | On the electrostatic interaction across a sakt solution between two bodies bearing unequal charges[END_REF][START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF][START_REF] Ben-Yaakov | Beyond standard poissonboltzmann theory: ion-specific interactions in aqueous solutions[END_REF][START_REF] Adar | Osmotic pressure between arbitrarily charged planar surfaces: A revisited approach[END_REF]. We will apply this formalism to calculate hydrodynamic flows in the context of wetting and/or dewetting situations in another manuscript.

Note that our theory is not restricted to a particular Gibbs free energy. Regarding diffusioosmosis or wetting we introduced a mesoscopic theory which describe them consistently, and also consistently with the contact value theorem. The theory presented here could be extended by taking into account other relaxation processes, such as thermal diffusion or solute diffusion in a way analogous to the approach described in references [START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF][START_REF] Landau | Statistical Physics, part 2[END_REF].

VIII. APPENDIX A. Free energy

In order to be specific, let us consider here the most simple Gibbs free energy that is used for describing spinodal decomposition in polymer blends for instance [START_REF] Cahn | Free energy of a non-uniform system. ı. interfacial free energy[END_REF][START_REF] Safran | Statistical Thermodynamics of Surfaces, Interfaces, and Membranes[END_REF]. Note that our approach is more general than this however. We consider a polymer blend made of two different polymers A and B of same molecular weight. We assume that the system may be considered as incompressible. The system may therefore be described by the variable ψ = ψ A -ψA = 1 -ψ B -1 + ψB = -ψ B + ψB where ψ A and ψ B are the volume fraction of component A and B respectively. One may consider the Gibbs free energy of the blend, as calculated for instance in reference [START_REF] Cahn | Free energy of a non-uniform system. ı. interfacial free energy[END_REF][START_REF] Safran | Statistical Thermodynamics of Surfaces, Interfaces, and Membranes[END_REF]. The typical order of magnitude of the Gibbs free energy of non polar liquids per monomer is of order the thermal energy T . As a consequence,

B. Non-zero slipping length

We assume here that the interfacial perturbation due to the solid-liquid interface is non zero only at the interface. We assume also that the liquid slips at the interface, an effect characterized by a so-called slipping length b which is the extrapolated length at which the liquid velocity would be zero (see e.g. [START_REF] De Gennes | Capillarity and Wetting Phenomena. Drops, Bubbles, Pearls, Waves[END_REF][START_REF] Ajdari | Giant amplification of interfacially driven transport by hydrodynamic slip: Diffusio-osmosis and beyond[END_REF]). If the interface is located at z = 0, Γ as a 3D function may be written as Γ(ψ(r), z) = Γ(ψ(x, y, 0))δ(z) [START_REF] Julicher | Generic theory of colloidal transport[END_REF] where δ(z) is the Dirac function.

Let us assume that the gradient of concentration ψ is constant along the capillary as well as dΓ/dψ. By taking the derivative of Eq.20 with respect to x and by using the incompressibility condition and the invariance by translation of the other terms, we deduce that the pressure p satisfies to the equation ∂ 2 p ∂x 2 = 0. If the pressures at both extremities of the capillary are identical, e.g. the atmospheric pressure, we deduce from this equation that the pressure is constant and uniform within the capillary. This equation allows to calculate the flow velocity profile. One has ∂ 2 v x ∂z 2 = 0 when either z > 0 or z < 0 where the function Γ is zero. One obtains thus that ∂vx ∂z is constant either for z > 0 or for -b < z < 0. The constant for z > 0 is zero by symmetry in a capillary. The shear rate for z < 0 is constant also. By solving Equation 20 This result had been proposed long ago by Ruckenstein [START_REF] Ruckenstein | Can phoretic motions be treated as interfacial tension gradient driven phenomena ?[END_REF], without the introduction of a Gibbs free energy and a virtual work principle. We obtain that the diffusio-osmosis mobility is proportional to the slipping length and to the derivative of the liquid-solid surface tension as regard to the solute concentration. -∇p(r) + ∇.η ∇v(r) + ∇v + (r) -ψ(r)∇µ tot (r) = 0 (29)

∇.v = 0 instead of Eq.18. The total chemical potential is given by Eq.19. Equation 30 differs by a term ∇ (ψµ tot ) from the Stokes equation Eq.18. They are equivalent if there is no contribution at the boundaries which is the case if one considers a 3D fluid problem with periodic boundary conditions for which both equations describe the same physics. But they are not equivalent in general. Apart from the definition of the chemical potential (in particular the term -ψ(r)∇ ∂Γ(ψ,r) ∂ψ ) this equation is similar to that discussed in references [START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF][START_REF] Liu | Pressure gradients failto predict diffusio-osmosis[END_REF][START_REF] Araki | Wetting-induced depletion interaction between particles in a phaseseparating liquid mixture[END_REF]. By focusing on the interfacial effects and neglecting µ = µ (0) + µ (1) , the Stokes equation 30 becomes:

-∇p + ∇.η ∇v + ∇v + -ψ∇ ∂Γ(ψ, r) ∂ψ = 0

∇.v = 0 ( 30
)
The velocity is zero in the z-direction. One obtains then -∂p(x, z) ∂z -ψ ∂ ∂Γ(ψ,(x,z)) ∂ψ ∂z = 0

By integrating between the current z-position and far from the solid surface, one obtains an expression for the pressure term :

p(x, z) = -ψ(x) ∂Γ(ψ(x), (x, z)) ∂ψ + p 0 which is the same as Equation 2if we identify ∂Γ(ψ(x),(x,z)) ∂ψ = Ψ(z). Equation30 becomes then: ∂ψ(x) ∂x ∂Γ(ψ(x), (x, z)) ∂ψ + η ∂ 2 v x ∂z 2 = 0 which is the same as Equation 3. This equation yields the same flow field as Equation 20but not the same pressure field. The pressure field p(x, z) cannot be the real physical pressure since it is not possible to adjust its value at the boundaries of the capillary: we have a difficulty with the boundary conditions. This approach is that of Derjaguin as reported in [START_REF] Anderson | Colloid transport by interfacial forces[END_REF][START_REF] Derjaguin | Kinetic phenomena in boundary films of liquids[END_REF][START_REF] Marbach | Osmotic and diffusio-osmotic flow generation at high solute concentration. i. mechanical approaches[END_REF]. Though the flow calculated likewise is the same as that calculated above, both points of view are difficult to reconcile. In any case, the pressure field obtained here contradicts the contact value theorem [START_REF] Israelachvili | Intermolecular and Surface Forces[END_REF].

IX. FOR TABLE OF CONTENT

vFIG. 1 :

 1 FIG.1: Diffusion-osmosis flow. The driving force in Eq.20 appears as a tangential force in the liquid distributed within a distance a of the solid surface. This force, though purely internal to the system, is analogous to the force in electro-osmosis which is exerted within the Debye length.

FIG. 2 :

 2 FIG.2:The driving force term in Eq.20 is non zero in the vicinity of the liquid A-liquid B interface, corresponding to the surface tension between both liquids, and on the triple line where the contribution of the solid surface takes place. This contribution is a tangential body force in the liquid over a region indicated here of spatial extension a in the direction normal to the substrate and ξ in the direction parallel to the substrate. When integrated over the volume, it yields γ B -γ A .

  y, 0) dψ when -b < z < 0. By applying the no-slip boundary condition at z = -b, that is v x (z = -b) = 0, one obtains the velocity profile: when -b < z < 0. Equations 27 and 28 represent a plug-flow.The diffusio-osmotic velocity in the capillary is then given by the diffusio-osmotic mobility defined by V = µ∇c is given by

C.

  Araki and Tanaka Stokes equation The Stokes equation is obtained by calculating the driving forces by using a principle of virtual work. A different Stokes equation as compared to Equation 18 may be obtained by integrating Equation 7 by part and by forgetting the boundary contributions. If we integrate also by part the contribution of the interface to the variation of the Gibbs energy given by Equation 17 and again do not consider the contributions at the boundaries, one obtains the Stokes equation :

FIG. 3 :

 3 FIG.3: Gradients of composition give rise to tangential forces which are responsible for diffusio-osmosis or wetting dynamics.

We have assumed a linear relation for γ as a function of ψ, which is consitent with the assumption made when deriving Eq.20. If this relation does not hold, one may modify Eq.20 without making this assumption. One obtains that the pulling force is indeed γ B -γ A

G/T is of order one per monomer volume a 3 . If one chooses for unit length scale the monomer length, the dimensionless Gibbs free energy may be written as G/T = d 3 rg(ψ) where the length scale has been made dimensionless and g is dimensionless of typical order of magnitude 1. A particularly frequent expression for the Gibbs free energy is given by

where the coefficients in the free energy term are all of order 1, except τ . τ which controls the miscibility of the two components may vary between a very low value, such as 10 -3

(weak segregation) or 1 (strong segregation). This expression for the Gibbs free energy is valid close to the critical point. It may be extended by taking higher order terms in the expansion in powers of ψ. The surface tension A-rich and B-rich domains is Γ ≈ 1/ξ 3 , and the correlation length or the thickness of the interfacial regions is ξ = τ -1/2 (assuming K = 1 so that Γ ≈ τ 3/2 . In physical units, the surface tension is then Γ ∼ T τ 3/2 /a 2 and is thus of order T /a 2 ∼ 2 × 10 -2 J m -2 in the case of strong segregation. In physical units, the correlation length is thus ξ ∼ aτ -1/2 . The corresponding dimensionless chemical potential (including gradient terms) reads

where ∆ is the Laplacian operator. The surface tension is entirely controlled by non-local contribution to the Gibbs free energy and is proportional to the parameter K which is typically proportional to T a 2 where T is the thermal energy and a one monomer length.

This dimensionless free energy obtained by dividing the real free energy by T and by choosing as unit length the length of one monomer is a generic free energy able to describe non-polar liquid blends close to the critical point, but it is by no means the only possibility. It may be useful to include higher order terms in the expansion in order to better control the fluctuations of ψ and in particular to prevent negative values in numerical calculations. This is particularly required when describing systems in the strong segregation limit where teh A-rich regions contain a very small fraction of B-polymer and the B-rich regions a very small fraction of A polymer.