Chen Yong-Gui

Cui Yu-Jun
email: yujun.cui@enpc.fr

Anh Tang

Minh

Wang Qiong

Y E Wei- Min

Prof Yu-Jun Cui

A preliminary study on hydraulic resistance of bentonite/host-rock seal interface

Keywords: Clays, expansive soils, laboratory test, radioactive waste disposal

Compacted bentonite-based materials are often used as buffer materials in radioactive waste disposal. When the compacted bentonite blocks are emplaced, technological voids related to different interfaces involving the buffer material are created, and their hydro-mechanical behavour is of primary importance for the disposal safety. In this study, the hydraulic resistance of the interface between compacted MX80 bentonite and Boom Clay was investigated in the laboratory using an injection cell. The results obtained show that when water is injected, the technological gap is quickly reduced due to the bentonite swelling. When water pressure reached the hydraulic resistance of the interface, the hydraulic fracturing took place with a drastic pressure decrease.

After fracturing, water injection continued and bentonite continued to swell. A higher subsequent pressure was needed to produce a new hydraulic fracturing. After a certain time, the hydraulic resistance becoming high enough no further fracturing occured, suggesting that the technological gap was sealed.

INTRODUCTION

To safely dispose high-level radioactive waste (HLW), the deep geological repository concept has been adopted in several countries. For such repository at great depth, the buffer material employed to seal the waste canisters must have high swelling potential, low permeability and good adsorption capacity. Blocks made of compacted bentonitebased material are usually considered for this purpose.

When bentonite bricks are placed around the waste canisters in the sealing buffers, some technological voids either between the bricks themselves or between the bricks, the canisters and the host rock are created. For instance, 10 mm thick gaps between the bentonite blocks and canister and 25 mm thick gaps between the bentonite blocks and the host rock have been considered in the basic design of Finland (Juvankoski, 2010). The joint gaps are limited to 6.6% of the total volume of the confining structure in the FEBEX mock-up test (Martin et al., 2006). In France, the volume of the bentonite/rock gaps is estimated at 9% of the volume of the gallery by the French waste management agency [START_REF] Andra | Référentiel des matériaux d'un stockage de déchets à haute activité et à vie longue -Tome 4: Les matériaux à base d'argilites excavées et remaniées[END_REF], while this value reaches 14 % in the Tournemire Underground Research Laboratory (URL) site [START_REF] Barnichon | Sealing experiments at the Tournemire URL, Towards convergence of technical nuclear safety practices in Europe[END_REF].

Once placed in the galleries, engineering barriers are progressively hydrated by pore water infiltration from the host rock. This water infiltration is strongly dependent on the initial state of the compacted material [START_REF] Cui | Determining the unsaturated hydraulic conductivity of a compacted sand-bentonite mixture under constant-volume and free-swell conditions[END_REF] and the imposed boundary conditions in terms of volume change (Yahia-Aissa et al., 2001). Thereby, the swell allowed by the technological voids described above has a significant influence on the hydro-mechanical behavior of the compacted bentonite. Indeed, swelling results in the decrease of dry density that may lead to the degradation of the hydro-mechanical performance of engineering barriers [START_REF] Komine | Self-sealing capability of some bentonite buffers in artificial seawater[END_REF]Komine, 2010). Although the compacted bentonite swelling fills the technological gaps, the filled gaps still represent discontinuities over long time. As a result, the safety function expected in the design may not be properly ensured. Thereby, a good understanding of the interface gap behaviour under water pressure is essential.

In this study, three laboratory tests were conducted to evaluate the behaviour of the interface between compacted MX-80 bentonite and natural Boom Clay with different initial gaps. It is expected to provide useful information for the interface behaviour between bentonite blocks and host rock in the Praclay heating test conducted in the URL at Mol, Belgium.

MATERIALS AND METHODS

The physical properties of MX80 bentonite and Boom Clay are shown in Table 1.

A compacted MX80 bentonite block as those used in the Praclay heating test was employed with the dimensions shown in Figure 1. Its dry unit mass is 1.80 Mg/m 3 , and its water content is 15.2%, defining a degree of saturation of 78.4%. The samples were prepared by coring from the block, and have a diameter of 50 mm or 55 mm and a height of 40 mm. Boom Clay cores taken from the Mol URL were used. The cores have a dry unit mass of 1.65-1.71 Mg/m 3 , a water content of 21-25% and a degree of saturation of 91-100%. For the sample preparation, the core was first cut into small cylindrical pieces of 50 mm high and 100 mm diameter each using a metal saw. A mould was used to fix the cylindrical pieces, and a machine was employed to prepare a hole of 60 mm diameter. Afterwards, the two ends of the hollow cylinder sample was cut to have reach 40 mm height.

The experimental set-up is shown in Figure 2. The sample is placed inside a cell that is put in a rigid frame with a load transducer of 50 kN capacity that allows the measurement of axial stress. The water inlet at the bottom was connected to a controller of pressure/volume (CPV) for water injection. Inside the cell, a hollow cylinder of Boom Clay and a cylindrical bentonite specimen were installed. The space between the bentonite and Boom Clay samples allowed the technological gap to be simulated. The height of the samples was kept constant by blocking the piston.

A miniature total pressure sensor of 8 mm diameter and 2 mm thick was selected to monitor the radial pressure at the Boom Clay/bentonite interface (see Figure 3).

Firstly, a small flat-bottomed hole with groove was prepared at the internal surface of Boom clay. Then the sensor was introduced into the hole and its cables were put into the groove. The scaning surface of the sensor was kept flush with the inner surface of Boom Clay. After the bentonite specimen was placed in the centre of the hollow cylinder of Boom Clay, the cables were connected to the data acquisition system.

When water was introduced to the cell, the compacted bentonite was hydrated and radially swelled. The sensor recorded the generated pressure when the bentonite contacted Boom Clay.

Three tests with two different technological gaps were carried out on samples with the same initial water content and initial dry density (see Table 2). An initial axial stress of 0.5 MPa was applied on the specimen before hydration to ensure a good contact between the piston and the sample (see Figure . 2). When water injection started, the piston was fixed. In each test, the sample was hydrated by injecting distilled water under constant pressure (0.1 MPa) through a porous disk in contact with the bottom face, while the top face was put in contact with another porous stone so as to allow free expulsion of either air or water (see Figure 2). The outlet was closed when water flowed out of it. Once water was observed on the top of the cell, the rate of water injection was fixed at 1 mm 3 /s. Changes in axial stress, radial pressures and injected water volume were monitored. When water pressure reached a certain value under which hydraulic fracturing occurred at the bentonite/Boom Clay interface, the water pressure decreased drastically. After the fracturing, the water injection was reset to the initial rate of 1 mm 3 /s until the next fracturing. This operation was repeated until no further hydraulic fracturing occurred. The dry density and water content of the soil specimen were determined at the end of the test.

RESULTS AND DISCUSSION

Figure 4-6 show the changes in pressure, volume of water injected, axial and/or radial pressure. For all cases, when the injection pressure exceeded a certain level, a sudden decrease took place. This sharp decrease corresponds to the phenomenon of hydraulic fracturing, and this pressure level is called herein breakthrough pressure or hydraulic fracturing pressure.

Further examination shows that after starting water injection, the injection pressure started increasing after 1-2 h. In Test A, the first fracturing was observed at t = 15 h with a sharp pressure decrease from 2.9 MPa to 2.2 MPa. Afterwards, various hydraulic fracturing occured with an increase of the hydraulic resistance each time.

When water was injected into the cell, it appeared to flow freely through the gap at the beginning. The water pressure did not increase before the gap was sealed by hydration that led the compacted bentonite to swell. Once the compacted bentonite was put in contact with water, it swelled and consequently reduced the gap. This sealing slowed down the water flow, resulting in an increase of the water pressure needed to keep a constant injection rate controlled by the CPV. When the water pressure reached the hydraulic resistance of the interface, hydraulic fracturing took place, resulting in a drastic pressure decrease. After fracturing, water was continued to be injected into the cell and bentonite continued to swell, thus, the hydraulic resistance was increased, resulting in a higher hydraulic fracturing pressure.

Comparing the injection pressure in Test A to those on the two other tests, it appears clearly that the pressure in Test A increased more quickly than in Tests B and C due to the smaller gaps.

As for the axial pressure, it increased quickly just after starting the injection in relation to the bentonite swelling. When the fracturing was observed, the axial pressure also abruptly decreased. The hydraulic fracturings identified through the changes in injection pressure are thereby observed again. The trend of the axial pressuretime curves shows that the axial pressure in Test A with 2.5 mm gap increased more quickly than that in Tests B and C with 5 mm gap. In particular, once the injection was stopped in Test A, the axial pressure also dropped and a steady final pressure induced by hydration was observed. However, in Tests B and C, some mud was observed escaping from the top of the cell during the injection. At that moment, the axial pressure fluctuated following a similar trend as injection pressure , then they both decreased to zero. It is supposed that the larger gap resulted in a larger radial strain (10 mm/50 mm: 20%), and this deformation led to collapse of the compacted bentonite. As a result, the axial pressure abruptly decreased to zero, the injection water outflowed with the collapsed soil, and the injection pressure did not increase any more over time.

On the whole, the changes in radial pressure are in accordance with those in injection pressure. When water was injected into the cell, the bentonite swelled and the gap was reduced. Once the bentonite was in contact with Boom Clay, the radial pressure increased progressively. The sealing of the gap slowed down the water flow, resulting in the increase of water pressure. When the water pressure reached the hydraulic resistance of interface, a fracturing occurred and resulted in a drastic pressure decrease. If the hydraulic fracturing took place, a sudden decrease of the radial pressure was also produced.

On the other hand, various hydraulic fracturings on the radial pressure were also observed with the increase of fracturing pressure. This is in agremment with the observation of [START_REF] Marcial | A laboratory study of the self-sealing behavior of a compacted sand-bentonite mixture[END_REF]. The increase of radial pressure applied by the swelling bentonite on the internal surface of Boom Clay evidenced a rapid swelling rate. Logically, a gap of 2.5 mm in Test A was sealed more quickly than a gap of 5 mm in Tests B and C. In Test A, it was observed that the radial pressure reached 1.8 MPa when the water pressure was controlled at 0.2 MPa. This indicates that the hydration process was continued even though the water pressure was kept constant. [START_REF] Komine | A Trial Design of Buffer Materials from the Viewpoint of Self-sealing[END_REF] reported that a necessary condition to demonstrate the selfsealing capability of buffer material is that the swelling pressure must be greater than 1 MPa once all voids are filled. Consequently, the interface between the Boom Clay and bentonite has enough swell capacity to seal a technical gap of 2.5 mm.

The distribution of water content and dry density of the soil samples after the tests are shown in Table 3. It appears that for each test the water content at the bottom was higher than that at the top. Moreover, the part near the interface was wetter than the other part. As for Boom Clay, the water content at the bottom was higher than that at the top, but the part near the cell was wetter than that near the gap. On the other hand, the dry density at the bottom and near the interface was lower than that at the top or far from the interface. This distribution is in good agreement with the wetting path.

The difference between the results of Test A and Test B and C can be explained by the difference in gap thickness. In Test A, the diameter of the compacted bentonite was 55 mm and the gap was 2.5 mm, less bentonite was needed to fill the gap, and thereby a higher final dry density was obtained. Refering to Tests B and C, the difference in distribution might be due to the different volume of water injected and the different test durations. The fact that the water content and dry density were not uniform in the bentonite shows that the swelling of bentonite was not homogeneous.

CONCLUSION

A simple experimental set-up was developed allowing investigation of the behaviour of the interface between Boom Clay and compacted MX-80 bentonite in terms of resistance to hydraulic fracturing. The obtained results allow the following conclusions to be drawn.

i) The water pressure did not increase at the first several hours after the start of injection until the gap was sealed by the compacted bentonite swelling. The axial pressure increased quickly in relation to the bentonite swelling, and it abruptly decreased when fracturing occured.

ii) When the gap was sealed, the radial pressure increased progressively. A sudden decrease of the radial pressure took place when hydraulic fracturing occurred. The fracturing pressure was found increasing over time.

iii) With a 2.5 mm gap, it was observed that fracturing pressure can be as high as 1.8 MPa that can be considered as high enough in the case of Boom Clay/compacted bentonite interface.

ii) The distribution of the water content and dry density of the soil after test depends on wetting path, gap thickness, injected water volume and elapsed time. r: the distance from the centre of the specimen to the sampling point; r 1 =7.5mm, r 2 =22.5mm, r 3 =35mm, r 4 =45mm.

  Figure 2. Schematic layout of the experimental set-up

Table 1 .

 1 Mineralogical composition of the soils tested

		MX80 bentonite	Boom clay
		(Wang et al. 2013)	(Francois et al. 2009)
	Montmorillonite (%)	80	10
	Quartz (%)		60
	w l (%)	575	59-83
	w p (%)	53	22-28
	ρ s (Mg/m 3 )	2.77	2.67

Table 3 .

 3 Water content and dry density of the soil specimens after the tests

			w (%)			ρ d (Mg/m 3 )		Test
										duration
		r 1	r 2	r 3	r 4	r 1	r 2	r 3	r 4	(hours)
	Test A Top	35	35.3	26.2 25.5	1.32 1.32 1.55 1.58
	Center 35.1	39.0	26.4 26.3	1.54 1.30 1.54 1.57	680
	Bottom 40.1	42.2	28.6 28.2	1.09 1.24 1.48 1.48
	Test B Top	37.1	44.3	37.0 29.8	1.25 1.21 1.39 1.48
	Center 29.9	41.3	33.3 29.3	1.28 1.26 1.39 1.49	65
	Bottom 96.4 124.5	69.8 44.8	0.73 0.61 0.91 1.11
	`Test C Top	43.2	45.5	28.5 29.1	1.34 1.22 1.50 1.47
	Center 45.2	47.8	29.2 30.5	1.18 1.14 1.47 1.47	93
	Bottom 51.1	58.8	32.6 33.6	1.08 0.97 1.39 1.36

ACKNOWLEDGEMENTS ONDRAF/NIRAS (The Belgian Agency for Radioactive Waste and Enriched Fissile Materials) is gratefully acknowledged for its financial support. The authors are also grateful to the National Natural Science Foundation of China (No 41272287, 41030748), Innovation Program of Shanghai Municipal Education Commission (12ZZ032), Pujiang Program of Shanghai (13PJD029), and Fundamental Research Funds for the Central Universities.


Wei-Kang Song

Prof Yu -Jun Cui
email: yujun.cui@enpc.fr

Anh Minh Tang

Wen-Qi Ding

Development of a Large-Scale Environmental Chamber for Investigating Soil Water Evaporation

Keywords: environmental chamber, Fontainebleau sand, soil water evaporation, soil suction at surface

A large-scale environmental chamber was developed to study the soil water evaporation mechanisms. A large soil specimen (300 mm high, 800 mm large and 1000 mm long) was used, allowing sensors to be installed with minimal effect on the soil hydraulic properties. Sensors for measuring soil suction, temperature and volumetric water content were either buried inside the soil specimen or installed on the chamber's wall at various locations.

Other sensors for monitoring air temperature, relative humidity, air flow rate and soil surface temperature were installed at different locations above the soil surface. Meanwhile, various atmosphere conditions were controlled by an air supply system and a steady water table at the bottom of soil was set through a big water tank. Fontainebleau sand was studied and it was compacted in the chamber in layers. After saturation, an 11.5-day evaporation test was performed. The results obtained were presented in terms of evolutions of suction, volumetric water content, air relative humidity and soil/air temperature. The data of air relative humidity and air temperature were further used for determining the actual evaporation rate; the data of soil volumetric water content and soil suction were used for determining the soil water retention curve. The quality of the results obtained showed the performance of the environmental chamber developed. In addition, these results can be further analyzed for theoretical and numerical developments involving soil water evaporation.

Introduction

Soil water evaporation is an essential component in the land surface energy balance [START_REF] Daamen | Measurement of Evaporation from Bare Soil and its Estimation Using Surface Resistance[END_REF]. Water is lost from soil during evaporation, thereby influencing the soil behavior, especially for clayey soils. In the field of agronomy, water loss due to evaporation can significantly affect crops in the planting and germination periods [START_REF] Lal | Principles of Soil Physics[END_REF]; in the geotechnical field, considerable water loss (e.g., drought) can induce significant soil volume changes, thereby damaging buildings and other infrastructures (e.g., [START_REF] Cui | Water Balance and Evapotranspiration Monitoring in Geotechnical and Geoenvironmental Engineering[END_REF][START_REF] Corti | Simulating Past Droughts and Associated Building Damages in France[END_REF][START_REF] Corti | Drought-Induced Building Damages from Simulations at Regional Scale[END_REF]. Furthermore, soil water evaporation also affects the design, selection and assessment of soil cover for landfill and mining application (e.g., [START_REF] Wilson | Coupled Soil-Atmosphere Modelling for Soil Evaporation[END_REF][START_REF] Yanful | Measurement of Evaporation Fluxes from Candidate Cover Soils[END_REF][START_REF] Yanful | Modeling and Measurement of Evaporation in Moisture-Retaining Soil Covers[END_REF]. Therefore, it is important to better understand the soil water loss process during evaporation.

In order to estimate soil water evaporation, various devices and methods have been developed.

Evaporation pan is usually used in the field for the measurement of free water evaporation that is considered as potential soil water evaporation (e.g., [START_REF] Blight | Interactions Between the Atmosphere and the Earth[END_REF][START_REF] Singh | Evaluation and Generalization of 13 Mass-Transfer Equations for Determining Free Water Evaporation[END_REF][START_REF] Fu | Investigating the Conversion Coefficients for Free Water Surface Evaporation of Different Evaporation Pans[END_REF][START_REF] Fu | A Critical Overview of Pan Evaporation Trends over the Last 50 Years[END_REF][START_REF] Li | Study of Desiccation Crack Initiation and Development at Ground Surface[END_REF]. For soil water evaporation investigation, several un-complex devices have also been developed. For instance, a circular pan with 300 mm in diameter but different heights and filled with compacted soil was developed by [START_REF] Kondo | A Parameterization of Evaporation from Bare Soil Surfaces[END_REF][START_REF] Kondo | A Model and Experimental Study of Evaporation from Bare-Soil Surfaces[END_REF], a soil column drying test device [START_REF] Wilson | Coupled Soil-Atmosphere Modelling for Soil Evaporation[END_REF]) and a pan with 258 mm in diameter and 74 mm in height and filled with thin enough soil sample by [START_REF] Wilson | The Effect of Soil Suction on Evaporative Fluxes from Soil Surfaces[END_REF]. The evaporation rate was obtained directly by weighing these devices over time.

Furthermore, the devices developed by [START_REF] Kondo | A Model and Experimental Study of Evaporation from Bare-Soil Surfaces[END_REF] and [START_REF] Wilson | Coupled Soil-Atmosphere Modelling for Soil Evaporation[END_REF] allow continuously monitoring the soil temperatures at various depths. For the water content profile, it was obtained only once by oven-drying at the end of experiment conducted by [START_REF] Kondo | A Model and Experimental Study of Evaporation from Bare-Soil Surfaces[END_REF] while the evolution of it during evaporation can be obtained by direct measurement via sampling ports at various depths in the test performed by [START_REF] Wilson | Coupled Soil-Atmosphere Modelling for Soil Evaporation[END_REF]. More recently, [START_REF] Mccartney | Effects of Infiltration and Evaporation on Geosynthetic Capillary Barrier Performance[END_REF] developed a large soil column evaporation system based on the infiltration test devices developed by [START_REF] Stormont | Capillary Barrier Effect from Underlying Coarser Soil Layer[END_REF], [START_REF] Bathurst | A Column Apparatus for Investigation of 1-D Unsaturated-Saturated Response of Sand-Geotextile Systems[END_REF] and [START_REF] Bathurst | Experimental Investigation of Infiltration Ponding in One-Dimensional Sand-Geotextile Columns[END_REF], to investigate both the evaporation and infiltration processes. Their system allows the simultaneous measurements of soil temperature, water content/suction. Following the similar measurement principle that water evaporation approximately equals the mass loss of soil, weighing lysimeter was developed that allows direct measurement of evaporation as changes in total mass of soil (e.g., [START_REF] Benson | Field Evaluation of Alternative Earthen Final Covers[END_REF][START_REF] Benli | Determination of Evapotranspiration and Basal Crop Coefficient of Alfalfa with a Weighing Lysimeter[END_REF]. In order to make the in-situ measurement simpler and more accurate, micro-lysimeter are used (e.g., [START_REF] Boast | A 'Micro-Lysimeter' Method for Determining Evaporation from Bare Soil: Description and Laboratory Evaluation[END_REF][START_REF] Plauborg | Evaporation from Bare Soil in a Temperate Humid Climate -Measurement Using Micro-Lysimeters and Time Domain Reflectometry[END_REF][START_REF] Wang | Measurement and Simulation of Evaporation from a Bare Soil[END_REF][START_REF] Qiu | An Improved Methodology to Measure Evaporation from Bare Soil Based on Comparison of Surface Temperature with a Dry Soil Surface[END_REF][START_REF] Bonachela | Measurement and Simulation of Evaporation from Soil in Olive Orchards[END_REF][START_REF] Liu | Determination of Daily Evaporation and Evapotranspiration of Winter Wheat and Maize by Large-Scale Weighing Lysimeter and Micro-Lysimeter[END_REF]. Micro-lysimeter can also be combined with water content sensors like TDR (time domain reflectometry) for water evaporation monitoring [START_REF] Wythers | Bare-Soil Evaporation Under Semiarid Field Conditions[END_REF].

A better control of atmospheric conditions is obviously essential in investigating soil water evaporation mechanisms. In this regard, the wind tunnel system is a good example. Typically, this system allows not only the control of wind velocity and solar radiation, but also the monitoring of air temperature and relative humidity (e.g., [START_REF] Yamanaka | A Modified Surface-Resistance Approach for Representing Bare-Soil Evaporation: Wind Tunnel Experiments Under Various Atmospheric Conditions[END_REF][START_REF] Komatsu | Toward a Robust Phenomenological Expression of Evaporation Efficiency for Unsaturated Soil Surfaces[END_REF][START_REF] Yamanaka | Effects of Gravel Mulch on Water Vapor Transfer Above and Below the Soil Surface[END_REF][START_REF] Yuge | Quantification of Soil Surface Evaporation Under Micro-Scale Advection in Drip-Irrigated Fields[END_REF][START_REF] Wang | Wind Tunnel Experiments on Bare Soil Evaporation[END_REF]). This system can be used in combination with the experimental devices mentioned above like pan (e.g., [START_REF] Komatsu | Toward a Robust Phenomenological Expression of Evaporation Efficiency for Unsaturated Soil Surfaces[END_REF]), soil tank (e.g., Wang 2006), weighing lysimester (e.g., [START_REF] Yamanaka | A Modified Surface-Resistance Approach for Representing Bare-Soil Evaporation: Wind Tunnel Experiments Under Various Atmospheric Conditions[END_REF][START_REF] Yamanaka | Effects of Gravel Mulch on Water Vapor Transfer Above and Below the Soil Surface[END_REF]) and micro-lysimeter (e.g., [START_REF] Yuge | Quantification of Soil Surface Evaporation Under Micro-Scale Advection in Drip-Irrigated Fields[END_REF]. Furthermore, if some sensors are used for soil temperature, suction and volumetric water content monitoring, this system allows a comprehensive monitoring of parameters for studying soil water evaporation (e.g., [START_REF] Yamanaka | A Modified Surface-Resistance Approach for Representing Bare-Soil Evaporation: Wind Tunnel Experiments Under Various Atmospheric Conditions[END_REF][START_REF] Yamanaka | Effects of Gravel Mulch on Water Vapor Transfer Above and Below the Soil Surface[END_REF]). Another commonly used system is the environmental chamber. A fast air circulation box was developed by [START_REF] Kohsiek | A Rapid-Circulation Evaporation Chamber for Measuring Bulk Stomatal Resistance[END_REF] with the simulation of wind. It is a useful chamber for the measurement of stomatal resistance of grass. After some minor adjustments and equipment of a fast dry and wet bulb thermocouple and a thermal infrared radiometer, this box was then used for soil surface resistance investigation (van de Griend and Owe 1994). Furthermore, based on the principle that changes in absolute humidity at inlet and outlet of the environmental chamber is due to the soil water evaporation, [START_REF] Mohamed | Solute Transport Through Unsaturated Soil Due to Evaporation[END_REF] developed a new chamber for predicting the solute transfer in unsaturated sand due to water evaporation, and [START_REF] Aluwihare | Measurement of Evaporation on Bare Soil and Estimating Surface Resistance[END_REF] developed a chamber to study the surface resistance of bare soil. On the whole, these chambers focus on the control atmosphere conditions, such as wind velocity, relative humidity, temperature etc., but rarely account for the soil parameters such as water content and suction. On the other hand, [START_REF] Yanful | Measurement of Evaporation Fluxes from Candidate Cover Soils[END_REF] performed an evaporation experiment on a compacted soil using cylindrical columns placed in an environmental chamber. This chamber can control air temperature and relative humidity and measure soil temperature and water content at different depths during evaporation. However, the soil mass, temperature and water content measurements should be done outside the chamber, the measurements being not instantaneous and continuous. [START_REF] Tang | Development of a Large-Scale Infiltration Tank for Determination of the Hydraulic Properties of Expansive Clays[END_REF] developed a large-scale infiltration tank allowing instantaneous monitoring of soil water content, temperature and suction during evaporation [START_REF] Ta | An Environmental Chamber for Studying the Soil-Atmosphere Interaction[END_REF][START_REF] Cui | Experimental and Numerical Investigation of Soil-Atmosphere Interaction[END_REF].

The various devices and methods mentioned above show that the large-scale environmental chamber is a good tool for investigating soil water evaporation in the laboratory. Compared to the wind tunnel system, the environmental chamber is less expensive and easier to operate but can provide rich data involving both air and soil parameters. Moreover, it has the same function as the combination of the wind tunnel and lysimeter. However, most existing environmental chambers only have a good performance in controlling air conditions, the soil being hardly taken into account (e.g., [START_REF] Kohsiek | A Rapid-Circulation Evaporation Chamber for Measuring Bulk Stomatal Resistance[END_REF][START_REF] Van De Griend | Bare Soil Surface Resistance to Evaporation by Vapor Diffusion Under Semiarid Conditions[END_REF]Aluwilhare and Watanabe 2003). In addition, the relationship between actual evaporation and soil suction or water content near the soil surface was rarely studied.

In this study, a large scale environmental chamber (1000 mm long, 800 mm wide and 895 mm high) was developed for studying soil water evaporation. An 11.5-day evaporation test was performed on the Fontainebleau sand. Various sensors were buried in the soil or installed on the wall of chamber at different depths, allowing monitoring of soil temperature, suction, volumetric water content, in addition to the air temperature and relative humidity. The recorded data were further analyzed to determine the actual evaporation rate and soil water retention curve. Note that this study focuses on soil water evaporation. To some extent, it corresponds to an extension of unsaturated soil mechanics.

Experimental Setup

The experimental setup consists of an environmental chamber, a wind supply unit, an air collection unit, a water supply unit and a data logging system. A sketch of this system is shown in Fig. 1. Schematic views of the environmental chamber are presented in Fig. 2 anda photograph is shown in Fig. 3. The chamber includes the main body, the ventilation part, the soil column part, the water drainage layer and an acrylic chamber cover of 8 mm thick.

The main body is an acrylic transparent chamber fixed on a base. The chamber consists of four acrylic plates mounted together by epoxy glue. The chamber has a wall of 20 mm thick, an internal width of 800 mm and an internal length of 1000 mm (Fig. 2(a)). Silicon glue was used to seal the joints in the four corners for preventing any leakage of air or water.

The soil column was prepared by compaction. The sensors measuring volumetric water content and soil temperature were installed at various depths during the compaction. The drainage layer was a compacted gravel (diameter: 2 -4 mm) layer of 15 mm thick and sandwiched between two layers of geotextile of 1 mm thick (Fig. 2(b)). Two outlets were prepared at the bottom of the drainage layer for soil saturation, drainage and water supply.

The details of the sensors used are presented in Table 1 and their locations are shown in Fig. 2(b). These sensors were installed at different monitoring points in both the soil column and air. The volumetric water content sensors, namely ThetaProbe, were buried at different depths (i.e., 25 mm, 40 mm, 55 mm, 125 mm and 225 mm below the soil surface). Four high-capacity tensiometers of 1.5 MPa working suction (Cui et al. 2008, Tang et al. 2010) were installed on two sides of the wall at various depths (i.e., 25 mm, 77 mm, 173 mm and 276 mm below the soil surface). One tensiometer was placed near the surface of soil (10 mm below the soil surface in order to ensure the good contact between the tensiometer and soil).

Six soil temperature sensors (PT1000) were set every 50 mm along the soil column. Moreover, an infrared thermometer was fixed at the cover to measure the soil surface temperature. Six T3111 transmitters were mounted inside and outside the chamber. Two of them were placed at the air inlet and outlet. For the other four sensors, one was fixed on the chamber's wall in the middle from the soil surface to the cover of chamber (i.e., 275-mm height); the second one was mounted outside the chamber for monitoring the laboratory relative humidity; the last two sensors were placed on the soil surface and at 50 mm above the soil surface, respectively.

The thermistors that allow the measurement of air temperature were fixed at different elevations along one side of the wall in the ventilation part.

The wind supply unit (see Fig. 1) was used for controlling the atmospheric conditions such as air temperature and air flow rate. This system consists of five parts: (1) high-pressure compressed air source; (2) air flow rate measurement unit; (3) air heating unit; (4) relative humidity and temperature measurement unit; and (5) air distributor. The compressed air source corresponds to the common laboratory compressed air system. The air flow rate is controlled by a valve and is monitored by a flowmeter. The air heating unit consists of heating hoses and temperature regulator. This unit can heat the air to a temperature up to 250 °C. The unit measuring air relative humidity and temperature is equipped with a rigid plastic cell in which a T3111 transmitter is inserted. The air distributor is a metal tube on which eight holes of 8.4 mm in diameter are drilled along the length of tube with a spacing of 100 mm.

The air collection unit, assembled on the opposite wall to the air distributor, is half of a polyvinyl chloride cubic box of 755 mm long, 30 mm large and 100 mm high. This unit collects the air from the chamber and a T3111 transmitter inside measures both the relative humidity and temperature of air. A total of five holes of 25 mm diameter in the wall of chamber enable the air to entre the collection unit.

The water supply unit for the chamber consists of a plastic water tank and a water table survey tube. The water tank supplies water to the chamber and the water level inside the tank is kept the same as the water table in the chamber (bottom of chamber in this study). The water table survey tube is a glass tube with marks and connected to the water tank. Thereby, any change of water table in the chamber can be detected. When the water table lowers down due to soil evaporation in the chamber, more water is added to the tank to keep a constant water table. The quantity of water added is also recorded.

Material and Experimental Procedure

The soil used for this experiment is the Fontainebleau sand. It is a natural, fine, white siliceous sand. Its specific gravity, maximum density and minimum density are 2.64, 1.39 Mg/m 3 , and 1.75 Mg/m 3 , respectively. The effective grain size D 10 is 0.14 and the coefficient of uniformity, C u = D 60 /D 10 , is 1.6 (see [START_REF] Delfosse-Ribay | Shear Modulus and Damping Ratio of Grouted Sand[END_REF]).

For the compaction of sand, 68 kg of dry sand was first poured into the tank and compacted manually to have a layer of 50 mm thick, corresponding to a dry density of 1.7 Mg/m 3 . This procedure of compaction was repeated until reaching the total height of 300 mm.

During compaction, the installation of sensors was performed. The PT1000 sensors measuring the soil temperature were buried above each layer (the spacing was then 50 mm). For the ThetaProbe sensors, two were inserted in the sand during the compaction and the others were buried in the first 60 mm below the soil surface after the soil saturation. For burying the ThetaProbe sensors, a hole having similar dimensions as the sensor was created manually at the defined level, and then the sensor was placed horizontally in the hole by inserting the four steel guides inside the soil. The hole was finally filled and manually compacted with a previously determined quantity of sand in order to ensure the same dry density. This procedure aimed at minimizing the effect of sensors installation on the soil density as described by [START_REF] Tang | Development of a Large-Scale Infiltration Tank for Determination of the Hydraulic Properties of Expansive Clays[END_REF].

After the soil compaction and sensors installation, the soil column was saturated through the water tank connected to the bottom of chamber. After saturation, the water level in the tank was lowered to a depth of 280 mm, i.e., the bottom of soil column. The installation of tensiometers was conducted during this water drainage progress. Furthermore, a relative humidity sensor on the soil surface was installed and the cover of chamber was sealed by silicon to ensure the air-tightness.

As far as the evaporation rate calculation is concerned, the basic principle is the calculation of the variation of air absolute humidity at the inlet and outlet of chamber. This method is described as follows [START_REF] Mohamed | Solute Transport Through Unsaturated Soil Due to Evaporation[END_REF][START_REF] Aluwihare | Measurement of Evaporation on Bare Soil and Estimating Surface Resistance[END_REF]:

The evaporation rate is calculated by the following expression:

_ _ ( ) 86400 a outlet a inlet a w Q H H E A    (1)
where E a is the actual evaporation rate (mm/day), H a_outlet is the absolute humidity at outlet (Mg/m 3 ), H a_inlet is the absolute humidity at inlet (Mg/m 3 ), Q is the air flow rate through the chamber (L/s),  w is the density of water (Mg/m 3 ) and A is the area of soil evaporation surface in the chamber (m 2 ).

The absolute humidity (H a ) is calculated as follows:

a a a a sat R R R R e t t t t     (4) 373.15 1 a R a t T  (5)
where e a is the vapor pressure (Pa); T a is the air temperature (K); R is the gas constant (287.04 J•kg -1 K -1 ); e sat is the saturated vapor pressure (Pa); H r is the air relative humidity (%); and 0.622 is the ratio of the molecular weights of water and dry air.

In this study, the air flow rate was maintained at 172 L/min and the heating tube temperature was 200 °C which corresponds to a temperature of 47 °C at the inlet of chamber.

Experimental Results

The air supply unit provided compressed hot air to the chamber at a rate of 172  5 L/min.

Figure 4 shows changes of air temperature over time. The values in the chamber increase during evaporation within a range from 24 °C to 32 °C. The shapes of the curves are similar showing a slight increase during the first six days and a quick increase during the last six days.

The values are very similar when the locations are above 185 mm. Note that the sensors at 275 mm, 380 mm and 465 mm above the soil surface give similar temperatures and they are therefore termed as "other sensors" in Fig. 4.

The changes of air temperatures at the inlet, outlet of chamber and in the laboratory are shown in Fig. 5. The value at the inlet is 47  3 °C, whereas the value at the outlet is lower and is increasing during the test from 25 °C to 30 °C. The laboratory room temperature varies from 20 °C to 24 °C and is lower than at the inlet and outlet.

The evolution of soil temperature is shown in Fig. 6. It is observed that the values increase slightly during the first six days but significantly during the last six days. The highest temperature is at the soil surface. In the deeper levels (25, 40, 55, 125 and 225-mm depths) corresponding to "other sensors" in this figure, the values are very close and increase from 18 °C to 25 °C. However, the soil surface temperature increases from 19 °C to 30 °C. Note that the surface temperature is not available for the period of t = 2 -3 days due to some technical problems.

All the temperature data recorded are used to plot the air-soil temperature profiles (Fig. 7).

For the air temperature, the highest value appears at the elevation corresponding to the location of air distributor (300 mm above the soil surface), the temperature in the zone close to the cover being lower due to the influence of laboratory room temperature. Regarding the soil temperature changes, a sharp temperature decrease can be observed in the near soil surface zone. The air temperature is significantly higher than the soil temperature. days. This means that from t = 2 days to t = 7 days, the water loss in the soil takes place only in the zone from the soil surface to 120-mm depth.

The evolution of soil matric suction is presented in Fig. 11. All the suction values at various locations are increasing with the water loss. Near the soil surface, the soil matric suction increases gradually from 14 kPa at t = 0 day to 46 kPa at t = 8 days. It increases quickly and reaches the limit of the sensor (i.e., 1.5 MPa) a few hours later. For the soil suction at other depths, the value at 77-mm depth is higher than at 173-mm depth but the difference is small.

The lowest suction is at 276-mm depth. The profile of soil suction is presented in Fig. 12. A clear and sharp suction gradient is observed in the zone from the soil surface to 77-mm depth.

This gradient is increasing over time: it increases from 0.13 kPa/mm at the beginning to 0.46 kPa/mm at t = 8 days.

The simultaneous measurement of suction and volumetric water content at various depths during the drying process allows determination of the soil water retention curve, as shown in Fig. 13. For each level of soil suction measurement (see Fig. 11), the corresponding volumetric water content is determined based on the volumetric water content profiles shown in Fig. 10(a); the volumetric water content at the soil surface is extrapolated by taking a constant water content gradient of 0.2 %/mm in the near surface zone. An air entry value of 7 kPa can be estimated in Fig. 13. It is also possible to use the model proposed by [START_REF] Fredlund | Equations for the Soil-Water Characteristic Curve[END_REF] to fit the water retention curve:

    m n r s r w a e ) / ( ln          (6)
where θ w is the volumetric water content (%); θ s is the volumetric water content in saturated state (θ s = 35.6%); θ r is the residual volumetric water content (θ r = 4.4%); ψ is the matric suction (kPa); e is the base of natural logarithm (e = 2.71828); a, n and m are fitting parameters. The fitting curve shown in Fig. 13 corresponds to a = 45.74, n = 1.9 and m = 15.2.

The actual evaporation rate determined following Eq 1 and the evolution of suction gradient between soil surface and 77-mm depth are plotted in Fig. 14. Regarding the evolution of evaporation rate, three phases can be identified: from t = 0 to t = 6 days, the rate decreases slightly from 2.3 mm/day to 2.0 mm/day; in the next 4 days, it decreases rapidly from 2.0 mm/day to 0.9 mm/day; after t = 10 days, the value decreases slowly, from 0.9 mm/day to 0.8 mm/day in 1.5 days. As far as the suction gradient is concerned, it increases slowly from the initiation of evaporation to t = 8 days, and then quickly reaches 4.8 kPa/mm at t = 8.5 days.

Interestingly, the high suction gradient corresponds to the significant decrease of evaporation ratio, indicating the increase of soil resistance to evaporation by suction increase.

As far as the cumulative evaporation is concerned, the calculation results of two different methods are presented in Fig. 15. Method 1 corresponds to direct calculation according to the actual evaporation rate -the results are shown in solid line. Method 2 corresponds to indirect determination by summing up the quantity of water infiltrated and the quantity from changes of volumetric water content -the results are plotted in dashed line. Note that the quantity of water infiltrated is calculated through the mass of water flowing out of the water tank divided by the soil evaporation surface, i.e., 1000 by 800 mm. The changes of volumetric water content are determined by considering the volumetric water content profiles shown in Fig. 10(a). The cumulative evaporation derived from Method 1 increases linearly over time but slows down after six days due to the decrease of evaporation rate. A total of 20.4 mm water is evaporated at the end of test. The cumulative quantity of water infiltrated increases following a linear function with time from the beginning to t = 3.7 days; it starts to slow down after four days. A total of 7.7 mm of water enter the chamber at the end of test. The cumulative quantity from changes of water content profiles increases during the test and reaches 24.5 mm at the end. It appears clearly that Method 2 gives higher cumulative evaporation than by Method 1, 32.2 mm against 20.4 mm.

Discussion

The environmental chamber system is a promising method for soil water evaporation investigation. As mentioned before, a fast air circulation box was developed by [START_REF] Kohsiek | A Rapid-Circulation Evaporation Chamber for Measuring Bulk Stomatal Resistance[END_REF] and a similar facility was used by van de [START_REF] Van De Griend | Bare Soil Surface Resistance to Evaporation by Vapor Diffusion Under Semiarid Conditions[END_REF], focusing on the reproduction of wind. Furthermore, the chamber built by [START_REF] Mohamed | Solute Transport Through Unsaturated Soil Due to Evaporation[END_REF] and the chamber system used by [START_REF] Aluwihare | Measurement of Evaporation on Bare Soil and Estimating Surface Resistance[END_REF] had a good control and measurement of air conditions but not the soil conditions. The environmental chamber presented in this study provides the possibility of simultaneous controlling/measuring both the atmospheric and soil conditions: the air conditions were controlled (see Fig. 4, Fig. 5 and Fig. 8) and soil parameters were monitored simultaneously (see Fig. 6, Fig. 7, Fig. 9 and Fig. 11).

In addition, the attempt of the suction measurement suction in the zone near soil surface (see Fig. 11) was also successful, which is, to the authors' knowledge, important and original results. On the other hand, this chamber has also the functions of the tunnel system developed by [START_REF] Yamanaka | A Modified Surface-Resistance Approach for Representing Bare-Soil Evaporation: Wind Tunnel Experiments Under Various Atmospheric Conditions[END_REF] and [START_REF] Yamanaka | Effects of Gravel Mulch on Water Vapor Transfer Above and Below the Soil Surface[END_REF].

As far as the thickness of the soil column is concerned, previous studies (e.g., [START_REF] Ta | An Environmental Chamber for Studying the Soil-Atmosphere Interaction[END_REF][START_REF] Cui | Experimental and Numerical Investigation of Soil-Atmosphere Interaction[END_REF] showed that only the zone close to the soil surface is subjected to the effect of evaporation. For this reason, the thickness of the soil column studied in the present work was reduced to 300 mm. Furthermore, the sensors used for volumetric water content and soil suction measurements were installed mainly in the near surface zone.

The soil water evaporation is an energy-consumption process. The main source of energy in this experiment is the hot air circulated above the soil surface: the constant air rate and the high inlet temperature (see Fig. 5) defined the energy for soil water evaporation. Therefore, at the beginning of evaporation (i.e., the first six days), the soil water evaporation consumed a lot of energy with a high evaporation rate (see Fig. 14), thereby, the air temperature increased at a low rate (see Fig. 4). With the decrease of evaporation rate, the energy consumed by water evaporation gradually decreased, resulting in air temperature increase at a high rate after six days. On the other hand, the difference between the inlet and outlet air temperature also shows that the soil water evaporation consumed energy from air. It should be noted that the high temperature of 200 °C at the heating pipe generated a temperature of 47 °C at the inlet and resulted in a temperature range from 24 °C to 32 °C in the chamber. This temperature range is quite usual in France [START_REF] Cui | Water Balance and Evapotranspiration Monitoring in Geotechnical and Geoenvironmental Engineering[END_REF].

The soil temperature change is an indicator of the energy (heat) change of soil during evaporation. Figure 6 shows that the soil temperature increased as the air temperature (see Fig. 4) was rising, indicating that the soil was heated by hot air and the energy for water evaporation was from the hot air. Furthermore, as evaporation is a progress of energy loss, the temperature gradient in the soil surface zone (25-mm depth) is larger than in other depths. This explains why evaporation is often limited to the near surface zone. Note that the soil surface temperature measured by infrared thermometer in this study is more accurate than that by sensors buried at the soil surface (e.g., [START_REF] Aluwihare | Measurement of Evaporation on Bare Soil and Estimating Surface Resistance[END_REF].

It can be noted in Fig. 7 that the air temperature in the zone close to the cover of chamber was slightly lower than in the middle-height of the air part. This phenomenon suggests that heat exchange existed between the environmental chamber and the laboratory ambiance. Therefore, when estimating the soil water evaporation in the chamber, the energy balance method can not be used. Regarding the fluctuation of soil temperature (about 0.5 °C) in deeper levels, it can also be attributed to the influence of ambient temperature. Indeed, the temperature sensors were buried at various distances from the chamber's wall (from 100 mm to 300 mm) and the laboratory temperature effect is expected to be different.

For the air relative humidity (see Fig. 8), its decrease inside the chamber suggests that the total water loss progressed (i.e., drying process) in the chamber. The relative humidity in the chamber was not affected by the ambient one, showing that the good performance of the chamber in controlling the air relative humidity; in other words, the ventilated part above the soil surface was sealed efficiently and the water evaporated from soil was completely transported to the outlet of chamber during the test. Thus, the calculation of evaporation based on the measurements of the temperature and relative humidity at the inlet and outlet is reliable.

The similar value of relative humidity at different positions of chamber (e.g., 50-mm height, 275-mm height and outlet) confirms the homogeneity of relative humidity in the chamber.

The large gap of relative humidity between the inlet and outlet shows the effect of evaporation in terms of supplying water vapor to the air.

The soil volumetric water content is an important indicator of water loss during evaporation.

The evolution of volumetric water content (see Fig. 9, Fig. 10(b)) shows a clear decline in the near surface zone (i.e., within 60-mm depth). This justifies the denser disposition adopted for the water content sensors in the near surface zone because it allows the water content profile to be well defined in this zone. The water content decline during the first six days corresponds to the initiation stage of evaporation (see Fig. 9, Fig. 10(a)) as reported by [START_REF] Wythers | Bare-Soil Evaporation Under Semiarid Field Conditions[END_REF]. In this stage, the evaporation rate is high and the quantity of water consumed is large. This explains the quick decline observed for the first six days. Afterwards, with the decrease of evaporation rate due to the increasing suction in the soil (see Fig. 14), the water content decrease was slowed down.

The measurement of matric suction near the soil surface using high-capacity tensiometer was successful. If the volumetric water content increased linearly with depth in the near surface zone (Fig. 10(a)), it is not the case for the suction which varied non-linearly with depth in this zone, as indicated by the water retention curve in Fig. 13. Thereby, the measurement of soil suction on soil surface is essential because we cannot estimate it by simple extrapolation. On the other side, soil suction on soil surface is a key parameter in the determination of soil water evaporation.

Regarding the evaporation rate (Fig. 14), it was decreasing during the test, showing three distinct evaporation stages: constant-rate stage (from t = 0 day to t = 6 days), falling-rate stage (from t = 6 days to t = 10 days) and low-rate stage (after t = 10 days), as also observed by [START_REF] Idso | The Three Stages of Drying of a Field Soil[END_REF] and [START_REF] Hillel | Introduction to Environmental Soil Physics[END_REF]. This evaporation process resulted from the following conditions:

(1) a continuous supply of heat by the hot air;

(2) a vapor pressure gradient between the soil surface and air -this gradient is reflected by the air relative humidity gradient above the soil surface (see Fig. 8);

(3) a continuous supply of water from the tank outside the chamber.

According to the water balance during soil water evaporation, the cumulative evaporation calculated by Method 1 should be equal to that by Method 2. However, Figure 15 shows that at the end of test, the cumulative evaporation calculated by Method 1 is less than Method 2: 20.4 mm against 32.2 mm. This could be attributed to the presence of trapped air in the gravel layer. During evaporation, the air bubbles dissipated and water could occupy the space initially occupied by air, leading thereby to water flow to the chamber from the water tank. In other words, this quantity of water just entered the gravel layer but not necessarily the soil layer. As a result, the cumulative evaporation from Method 1 is close to that from the cumulative changes of water content (i.e., 24.5 mm).

Conclusion

A large-scale environmental chamber was developed in order to study the soil water evaporation process. The atmospheric conditions (air rate, relative humidity and temperature)

were controlled and monitored. The soil column was instrumented by various sensors for measuring matric suction, volumetric water content and temperature. An evaporation test was performed on the Fontainebleau sand to verify the relevance of the setup developed. The following conclusions can be drawn:

The air temperature in the chamber was found to increase after six days, showing that with the decrease of evaporation rate due to the soil suction increase, the energy consumed was decreased, giving rise to air temperature increase.

The soil temperature was found to increase, indicating that the soil was heated by hot air and the energy for water evaporation was from the hot air. In addition, as evaporation is a progress of energy loss, the temperature gradient in the soil surface zone is larger than in deeper levels.

The temperature in the chamber was affected by the laboratory environment. But the relative humidity in the chamber was not affected by the relative humidity of the laboratory -This validated the method of actual evaporation determination based on the inlet and outlet relative humidity values.

The evolution of volumetric water content showed significant changes of water content in the near surface zone (within 60-mm depth). This justifies the denser disposition adopted for the water content sensors in the near surface zone on the one hand, and the choice of a limited height (300 mm) for the soil column on the other hand.

The attempt of near soil surface suction measurement by high-capacity tensiometer was successful. This measurement is, to the authors' knowledge, original results. These results are important since the soil surface suction is a key parameter in the determination of soil water evaporation.

The relevant data obtained allowed the determination of actual evaporation rate and the water retention curve. They also show the performance of the environmental chamber developed in studying soil water evaporation. Moreover, they can be used in further theoretical development for soil-atmosphere interaction investigation.

List of Tables

List of Figures

Furthermore, the temperature

  gradient above the soil surface (in the zone from 80 mm above the soil surface to the soil surface) decreases progressively over time while the gradient between the soil surface and 25-mm depth increases. The soil temperatures in deeper zone (from 25-mm depth to the soil bottom) are quite similar with a difference less than 0.5 °C.The changes of air relative humidity are shown in Fig.8. The values in the chamber decrease from 67.4 % to 23.8 % in the zone near the soil surface and from 35.4 % to 12.4 % at the outlet, while the values at the inlet are very low and nearly constant. On the whole, the variations of relative humidity (except that at the inlet of chamber and in the laboratory) can be divided into two parts: during the first six days, the relative humidity declines at a low rate; then it drops in the next six days. The value at the soil surface was higher than in other locations. The relative humidity in the laboratory shows a large fluctuation, from 19.7 % to 40.5 %, but this does not affect the values measured in the chamber.The changes of volumetric water content are shown in Fig.9. The volumetric water content at 60 mm below the soil surface decreases from 25.4 % to 7.1 % at 25-mm depth and from 31.9 % to 12.3 % at 55-mm depth. In the deeper locations, i.e., at 125 and 225-mm depths, there are no changes before t = 7 days. The value remains unchanged at 225-mm depth during the whole test while the value at 125-mm depth starts to change from t = 7 days. As far as the first 60-mm layer near the soil surface is concerned, the variation can be divided into two parts: at the beginning, the water content decreases quickly in the first six day and then decreases slowly and reaches a stabilization state at end of the test.The profiles of volumetric water content and the contour map are shown in Fig.10. The profiles in Fig.10(a) show a clear water loss process during evaporation. It can be noted that the gradient of water content between the three points at 25, 40, and 55-mm depths respectively is constant and equal to 0.2 %/mm. This gradient is also the maximum for the whole depth. Similar linear gradient of water content can be observed from 25-mm to 225-mm depth at end of the test (t = 11.5 days). The contour map allows the visualization of the drying front over time (see Fig.10(b)). For instance, the point having a water content of 30 % is at 50-mm depth at the beginning; this point goes down quickly and reaches the first stabilization stage at 120-mm depth after t = 2 days. It starts to increase again only at t = 7

FIG. 1 -

 1 FIG. 1-Sketch of the environmental chamber test system

TABLE 1 .

 1 The sensors used

Acknowledgments

The authors are grateful to the support of China Scholarship Council (CSC).


M E Hayden

E Baudin

G Tastevin

P J Nacher

NMR Time Reversal as a Probe of Incipient Turbulent Spin Dynamics

Keywords: numbers: 82.56.Jn, 75.50.Mm, 76.60.Jx, 76.60.Lz

We demonstrate time reversal of nuclear spin dynamics in highly magnetized dilute liquid 3 He-4 He mixtures through effective inversion of long-range dipolar interactions. These experiments, which involve using magic sandwich NMR pulse sequences to generate spin echoes, probe the spatiotemporal development of turbulent spin dynamics and promise to serve as a versatile tool for the study and control of dynamic magnetization instabilities. We also show that a repeated magic sandwich pulse sequence can be used to dynamically stabilize modes of nuclear precession that are otherwise intrinsically unstable. To date, we have extended the effective precession lifetimes of our magnetized samples by more than three orders of magnitude.

Elementary treatments of nuclear magnetic resonance (NMR) ignore collective effects. Individual spin dynamics are assumed to be independent of the sample magnetization M, and are thus governed by linear differential equations. This approximation is justified for many -but certainly not all -practical applications. One of the most well known and pervasive counter examples is the phenomenon of radiation damping [1], in which the emf induced in the pickup coil by M drives a current; the magnetic field associated with this current in turn acts on M, driving it into alignment with the static field B 0 in a nonlinear manner. A less common but more insidious problem arises when the magnetic field produced by the magnetized sample becomes large enough to directly influence spin dynamics [2,3,4,5,6]. This leads to a rich variety of nonlinear effects that range from spectral clustering to precession instabilities and spin turbulence. The former arises when small-angle NMR tipping pulses are applied to the sample (generating small angular displacements of M from B 0 ), and is manifest by the spontaneous appearance of long-lived geometrydependent modes of coherent nuclear precession. The latter occurs when large-angle tipping pulses are employed, and involves an exponential growth in the complexity of spatially inhomogeneous magnetization patterns. Far from being mere curiosities, phenomena arising from nonlocal interactions (including the joint action of 'distant dipolar fields' and radiation damping [7,8]) threaten to limit (or profoundly alter) the operation of state-of-the-art highfield high-resolution NMR spectrometers. They are equally important for understanding the dynamics of polarized quantum fluids including Bose-Einstein condensates [9], superfluid 3 He [10,11], degenerate 3 He-4 He mixtures [12], and two-dimensional H gases [13]. From yet another perspective, the collective effects induced by distant dipolar fields can be used to amplify weak spin precession signals [14] and to enhance the sensitivity of precision searches for CP-violating permanent electric dipole moments [5].

Here we describe NMR time reversal experiments that -for the first time -shed light on the extent to which the deleterious effects of spin turbulence can be controlled or suppressed. Moreover, they lay the foundation for a systematic and quantitative investigation of the onset of dynamical magnetization instabilities under diverse conditions. Our experiments are performed on dilute non-degenerate mixtures of hyperpolarized 3 He in liquid 4 He. They are based on the use of magic sandwich (MS) pulse sequences [15,16]: continuous bursts of tipping pulses that are appropriately timed and phased so as to refocus spin dynamics associated with dipolar interactions, leading to the formation of echoes. The MS pulse cycle and numerous variants are indispensable tools in the field of solid-state NMR but, to the best of our knowledge, have not been used previously for liquid-state NMR [17].

Our samples are confined to a slightly-prolate 0.44 cm 3 spheroidal Pyrex cell treated with Cs metal to suppress wall relaxation [START_REF]Wall contributions to the NMR relaxation times T1 and T2 are of order several hours under these conditions[END_REF][START_REF] Piegay | [END_REF], and are maintained at temperatures T ∼ 1 K. The sample cell communicates with a room temperature metastability-exchange optical pumping cell and various gas reservoirs via a narrow Pyrex tube. The entire apparatus is immersed in a 2.3 mT magnetic field shimmed to ±20 ppm over the sample cell and actively stabilized to ±10 ppm against fluctuations in the background laboratory field. We control the volume of liquid admitted to the cell (typically 40 % full), the concentration X of 3 He in the mixture (typically 1-5 %), and the nuclear polarization (typically 40 %). We can thus manipulate the shape of the sample, the initial amplitude of its magnetization M , and the 3 He diffusion coefficient D. Note that it is convenient to report M in terms of a characteristic dipolar field B dip = µ 0 M or dipolar frequency F dip = γB dip /2π, where µ 0 is the permeability of free space and γ is the 3 He gyromagnetic ratio.

Our pulsed NMR experiments are performed at 74 kHz using orthogonal transmit and receive coils. The transmit coils are actively-shielded [20] to suppress the induction of eddy currents in nearby conducting structures. This enables us to apply the intense and sustained rf (i.e. B 1 ) fields that are necessary for NMR time reversal experiments without causing the sample to warm. It also ensures that eddy currents do not distort the applied rf field, which is uniform to within ±200 ppm over the sample cell. The receive coils are tuned in a tank configuration with a quality factor Q 0 ∼ 10. The filling factor η is of order 0.02 when the cell is completely full, and an externally adjustable 'Q-spoiling' resistor is used to ensure that radiation damping fields are small compared to those generated by the magnetized sample. Typically a value Q ∼ 2 was employed for the work presented here. Figure 1 summarizes data from two experiments that illustrate some of the unusual behavior exhibited by highly magnetized liquids. The first experiment involves application of a single 90 • rf tipping pulse. The corresponding free induction decay (FID) is shown as trace (a). The signal amplitude S is roughly constant for several tens of ms, during which time the precessing transverse nuclear magnetization is necessarily uniform. This is followed by a precipitous drop that culminates in a number of small revivals. Similar experiments performed over a range of initial conditions indicate that the characteristic time scale for the abrupt decrease in signal amplitude (i.e. T 1/2 ) is inversely proportional to B dip [21].

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Evidence that this behavior is not dominated by terms in the equation of motion linear in M is provided by the second experiment, in which the initial 90 • pulse is followed 70 ms later by a 180 • pulse. Comparison of traces (a) and (b) reveals that the 180 • pulse has little if any effect on the FID envelope. In particular, there is no obvious sign of a Hahn (or inhomogeneous) spin echo 140 ms after the start of the experiment. Closer examination of these data shows that the deviation of the FID amplitude S from its initial value S 0 grows exponentially with time t (inset to Fig. 1): i.e. S 0 -S ∼ exp(2Γt). This behavior is expected for an infinite spin system dominated by distant dipolar interactions [6], and has been interpreted as the onset of spin turbulence [3]. We have performed exact numerical simulations of these experiments for small numbers of dipolar-coupled spins diffusing on a cubic lattice (up to 32 × 32 × 32 sites) with periodic boundary conditions [21], and observe excellent agreement between measured and calculated growth rates. In both cases Γ/F dip ∼ 2.3, which is somewhat smaller than the value 2 √ 2π/3 ∼ 3.0 predicted for an infinite spin system [6].

The experimental challenge presented by these data is to devise a method for probing the spatial and temporal evolution of the nuclear magnetization before, during, and after the point at which it spontaneously and abruptly averages to zero in Fig. 1. The approach we explore here as a first step toward this goal is the application of MS pulse sequences to reverse the effective time evolution of dynamics associated with dipolar interactions, and thereby attempt to generate spin echoes. Thorough descriptions of the manner in which time reversal is accomplished can be found in the literature [15]. For our purposes the essential features of these multiple-pulse sequences are that (a) the applied rf field must be large compared to the dipolar fields and (b) the effective rate at which the dipolar-coupled system evolves in time as the rf fields are applied is scaled by a factor of -1/2. FIG. 2: [color online] Spin echo formation: A 90 • pulse is applied at t = 0, after which the magnetization is allowed to evolve for τ = 70 ms. MS sequences of various durations are then used to progressively refocus the magnetization. For traces (a) -(c), the amplitude of the FID immediately after the MS probes the rotary echo formed in the doubly-rotating frame. The symbols , •, indicate pairings between points on the initial FID to which the state of the magnetization is nominally returned and the ensuing free precession signal. For traces (d) -(f), the amplitude of the FID probes the magic sandwich echo formed in the rotating frame after the magnetization is first 'unwound' back to t = -τ before being 'released' and allowed to undergo further free evolution. These data were acquired under conditions similar to those outlined in Fig. 1. Inset: A classic magic sandwich [15] consists of a continuous train of ( 180 • x , -180 • x ) rotation pairs sandwiched between a leading 90 • y and a trailing -90 • y rotation. We have used 360 • (rather than 180 • ) rotations to reduce the accumulation of phase errors associated with offsets between the rf and Larmor frequencies. Each 360 • rotation is accomplished in 2 ms, corresponding to B1 = 15 µT. Note that for trace (c) the sense of the terminal 90 • rotation was reversed (i.e. 90 • y instead of -90 • y ) in order to also refocus interactions with a linear dependence on M. This does not significantly alter the initial amplitude of the echo or its subsequent growth.

Figure 2 shows data from several experiments that are analogous to the ineffective 90 • -τ -180 • sequence of Fig. 1 except for the fact that the 180 • tipping pulse has been replaced with MS sequences of various durations. The rf used for these sequences is typically adjusted to be within 1 Hz of the 3 He Larmor frequency. The most striking and important feature of these data is that large NMR signals are detected long after the net transverse magnetization normally would have averaged to zero. The upper half of Fig. 2 shows data from three experiments that progressively 'unwind' the state of the dipolar-coupled system back to times t = 2τ /3, t = τ /3, and t = 0, as indicated by the paired-symbols , •, and . The initial amplitudes of these echoes nominally probe the amplitude of the 'rotary echo' [15,22] that is formed in the doubly-rotating frame [23] during the MS sequence. In the first case -trace (a) -the magnitude of the transverse magnetization is returned to a value that is close to that which is anticipated. Moreover, the shape of the ensuing FID mimics the original decay (cf. Fig. 1). In the last case -trace (c) -the amplitude of the signal following the MS starts out noticeably smaller than anticipated but then unexpectedly grows rather than decaying. Clearly, the deficit in the initial echo amplitude is not associated with an irreversible process. The shape of the FID labeled (b) is qualitatively intermediate between traces (a) and (c). The lower part of Fig. 2 shows data from three experiments that nominally 'unwind' the state of the dipolar-coupled system back to a time t = -τ , resulting in the formation of a 'magic sandwich echo' in the rotating frame [15] after a further free evolution period of duration τ . Increasing the initial magnetization increases the rate at which magnetization patterns develop, and so this series of experiments effectively probes the sample at successively later stages of evolution.

It is not surprising that we observe imperfect refocussing of the transverse nuclear magnetization in these experiments. Irreversible processes (e.g. diffusion) necessarily compete with the spatiotemporal evolution of the magnetization and progressively attenuate structures on larger and larger scales as time progresses. At the same time, while the MS tends to refocus the family of magnetization patterns that develop during the initial FID, one expects a complementary family of instabilities [6] to be established and grow during the time reversal sequence. Technical imperfections in the time reversal sequence will also contribute to echo attenuation. A detailed and systematic study of these rich and complex effects is required before quantitative statements can be made regarding the onset and development of spin turbulence.

The fact that spin echoes can be generated in highly magnetized liquids leads one to contemplate the use of more sophisticated NMR tools. To illustrate this point, we describe here a simple but significant extension of the basic MS sequence that clearly merits further investigation. A conventional CPMG sequence (90 • -τ -180 • -2τ -180 • -2τ . . .) acts on a system dominated by interactions that are linear in M, repeatedly inverting the phase of the precessing magnetization to produce an echo train. By analogy, a Repeated Magic Sandwich (RMS) sequence consisting of a train of MS cycles interspersed with appropriate delays should likewise act on a system dominated by distant dipolar interactions to form an echo train. The validity of this conjecture is borne out by the data shown in Fig. 3. The RMS sequence used to generate these data is of the form (90 • -τ -MS-2τ -MS-2τ . . .), where the duration of each MS cycle is 4τ (i.e. long enough to effectively 'reverse' the time evolution of the dipolar coupled system for a time 2τ ). The four traces correspond to experiments performed with different delay times τ . In each case, the RMS sequence is applied for ∼ 9 s, at which point M is allowed to evolve without further interruption. Each echo train starts with a period that we associate with imperfect refocussing of the magnetization. That is, behavior similar to the examples presented in Fig. 2. Increasing the length of time during which the magnetization is allowed to evolve between successive magic sandwiches results in greater losses. This initial period is followed by a second phase during which magnetization losses continue to be incurred, but at significantly lower rates. Comparison with Figs. 1 and 2 reveals that the apparent lifetimes of these dynamically sustained precession signals are in some cases more than 3 orders of magnitude longer than the characteristic coherence time (e.g. T 1/2 ) of the freely evolving magnetization. Exact numerical simulations of these experiments (analogous to those described above) have been used to investigate the influence of diffusion and rf field amplitude. They reveal similar -but not identical -behavior. The initial period during which losses are incurred is observed, as is the transition to a second evolutionary phase and the dependence of this transition on both τ and B dip . On the other hand, our model calculations yield a steady state solution for which we are not able to resolve a decay rate: that is, a plateau. This suggests that our experimentally-determined echo trains may yet be limited by systematic rather than intrinsic effects. One possibility is the small but finite heat load placed on our cryostat by eddy currents induced during RMS sequences, which would in turn lead to evaporation of 3 He atoms. Another possibility is imperfect refocussing of the magnetization caused by imperfect phasing of rf tipping pulses. Finally, it is worth commenting on the terminal FID at the end of each RMS sequence. Analysis of these decays yields values of Γ that are proportional to the terminal echo amplitude and consistent with those observed following a single 90 • tipping pulse (cf. Fig. 1). This strongly suggests that the precessing magnetization is highly uniform at the peak of each revival, and that the 'seeds' from which instabilities ultimately develop [6] are small.

In summary, we have demonstrated the production of NMR spin echoes and spin echo trains in magnetized liquids under conditions where nuclear spin dynamics (in the rotating frame) are dominated by nonlinear and nonlocal (or distant) dipolar interactions. Our experiments, which are based on the use of magic sandwich time reversal pulse sequences, probe the rich spatiotemporal evolution of the nuclear magnetization distribution that results from competition between nonlinear spin dynamics and diffusion. They promise to yield quantitative information about the progression and extent of disorder, as well as new methods to control or suppress the onset of instabilities.

MH gratefully acknowledges support from the CNRS (France), the ENS (Paris), and NSERC (Canada).

FIG. 1 :

 1 FIG. 1: [color online] FID response of polarized 3 He in liquid 4 He (a) to a 90 • rf tipping pulse and (b) to a 90 • -τ -180 • sequence. The time at which the FID amplitude S drops to half of its initial value S0 is denoted T1/2, which is of order 70 ms for these examples. Trace (b) has been displaced upward by 0.4 mV for the sake of clarity. The inset shows the deviation of trace (b)from its initial amplitude. These data were acquired at T = 1.14 K, with X ∼ 2 % and B dip ∼ 0.9 µT for trace (a) and X ∼ 5 % and B dip = 0.7 µT for trace (b).

FIG. 3 :

 3 FIG. 3: [color online] Spin echo trains: Traces (a) -(d) correspond to echo trains that are observed when RMS sequences with delay times τ = 16, 24, 32, and 40 ms, respectively, are applied to the magnetized liquid. The MS cycles used to obtain these data are identical to those described in the caption of Fig. 2 except for the facts that each one terminates with an additional 180 • y rotation (cf. trace (c) of Fig. 2) to refocus interactions linear in M, and that the symmetry of alternate MS cycles is inverted to reduce the accumulation of phase errors. That is, odd MS cycles are of the form [90 • y , n × (360 • x , -360 • x ), 90 • y ] while even MS cycles are of the form [-90 • y , n × (-360 • x , 360 • x ), -90 • y ]. These data were acquired at T ∼ 1.1 K, with X ∼ 4.5 % and B dip ∼ 0.8 µT. In the limit B dip → 0, CPMG-based measurements yield D ∼ 2 × 10 -7 m 2 /s.

Figure 3

 3 Figure3reveals two distinct phases in the evolution of the magnetization during the application of RMS sequences. Each echo train starts with a period that we associate with imperfect refocussing of the magnetization. That is, behavior similar to the examples presented in Fig.2. Increasing the length of time during which the magnetization is allowed to evolve between successive magic sandwiches results in greater losses. This initial period is followed by a second phase during which magnetization losses continue to be incurred, but at significantly lower rates. Comparison with Figs. 1 and 2 reveals that the apparent lifetimes of these dynamically sustained precession signals are in some cases more than 3 orders of magnitude longer than the characteristic coherence time (e.g. T 1/2 ) of the freely evolving magnetization. Exact numerical simulations of these experiments (analogous to those described above) have been used to investigate the influence of diffusion and rf field amplitude. They reveal similar -but not identical -behavior. The initial period during which losses are incurred is observed, as is the transition to a second evolutionary phase and the dependence of this transition on both τ and B dip . On the other hand, our model calculations yield a steady state solution for which we are not able to resolve a decay rate: that is, a plateau. This suggests that our experimentally-determined echo trains may yet be limited by systematic rather than intrinsic effects. One possibility is the small but finite heat load placed on our cryostat by eddy currents induced during RMS sequences, which would in turn lead to evaporation of 3 He atoms. Another possibility is imperfect refocussing of the magnetization caused by imperfect phasing of rf tipping pulses. Finally, it is worth commenting on the terminal FID at the end of each RMS sequence. Analysis of these decays yields values of Γ that are proportional to the terminal echo amplitude and consistent with those observed following a single 90 • tipping pulse (cf. Fig.1). This strongly suggests that the precessing magnetization is highly uniform at the peak of each revival, and that the 'seeds' from which instabilities ultimately develop[6] are small.In summary, we have demonstrated the production of NMR spin echoes and spin echo trains in magnetized liquids under conditions where nuclear spin dynamics (in the rotating frame) are dominated by nonlinear and nonlocal (or distant) dipolar interactions. Our experiments, which are based on the use of magic sandwich time reversal pulse sequences, probe the rich spatiotemporal evolution of the nuclear magnetization distribution that results from competition between nonlinear spin dynamics and diffusion. They promise to yield quantitative information about the progression and extent of disorder, as well as new methods to control or suppress the onset of instabilities.MH gratefully acknowledges support from the CNRS (France), the ENS (Paris), and NSERC (Canada).


Saurav Aryan
email: aryan002@umn.edu

Do vector plane waves form complete basis of solutions to Maxwell's equations? Introduction to Generalized Plane Wave Solutions

As the title says, this article questions the well established belief that the vector plane waves form a complete basis of solutions to Maxwell's equations. Vector solutions to Maxwell's equations are presented here, which have planar wave-fronts and transverse electric and magnetic fields but spatially varying polarization. They form a one-parameter family specified by integer n, and are

termed as generalized vector plane waves. The known vector plane wave solution with spatially uniform polarization, referred to as conventional vector plane waves in this article, is a subset of this family obtained for n = 0. In contradiction to the established belief, it is shown that these generalized vector plane waves with spatially varying polarization (for n = 0) cannot be expressed as superposition of conventional vector plane waves. The family of solutions also includes the interesting cases of radially and azimuthally polarized plane waves for n = 1.

I. INTRODUCTION

Vector Plane waves with polarization of uniform magnitude and direction are the simplest known solution to Maxwell's equations. Owing to mathematical complexity of full vector Maxwell's equations however, most problems in classical optics are treated with scalar wave equation. In Fourier optics, scalar plane waves are natural mode of propagation in free space, and any scalar electromagnetic field solution can be represented as superposition of plane waves, if the Fourier Transform of the solution exists. Analogously, it has been generally assumed in the scientific community that the vector plane waves form a complete basis for all solutions to vector Maxwell's equations. In this letter, a new class of vector solutions to Maxwell's equations is presented, which cannot be expressed as superposition of vector plane waves. The derivation is done without making any approximations and in vacuum region free of charge and current. These solutions form a one-parameter family specified by integer n, and termed as generalized vector plane waves since they possess planar wave-front but has spatially varying polarization, including radial and azimuthal polarization distributions. To avoid confusion, the known vector plane wave solution with spatially uniform polarization shall be referred as conventional vector plane waves whereas the new solution being presented in this paper shall be referred as generalized vector plane waves. Interestingly, the conventional vector plane wave solution itself is a special case of the family of generalized plane wave solutions, which is obtained for n = 0.

II. THEORY

The form of Maxwell equations in free space can be written as follows:

∇ • E = ρ 0 (1) ∇ • B = 0 (2) ∇ × E = -∂B ∂t (3)
∇ × B = µ 0 0 ∂E ∂t + J (4)
In a region free of charge and current, ρ = J = 0.

Copyright c 2018 Saurav Aryan. ALL RIGHTS RESERVED

The Maxwell's equations then reduce to wave equations in E under the Lorentz gauge condition [START_REF] Phillips Feynman | Mainly electromagnetism and matter[END_REF] ,

1 c 2 ∂ 2 E ∂t 2 -∇ 2 E = 0 (5)
where c = 1 √ µ 0 0 is the speed of light in vacuum.

Assuming time-harmonic form of the solution, we can write,

E(r, θ, z, t) = Ẽ(r, θ, z)e -iωt (6)
,

where ω is the frequency of the wave, and (r, θ, z) represent the cylindrical coordinate system. Substituting the above time harmonic electric field into (5), we obtain the Helmholtz equation in Ẽ

∇ 2 Ẽ + k 2 Ẽ = 0 (7)
where k = ω c is the wavenumber. As we would only focus on the time-independent part of the field, the tilde from Ẽ will be dropped in subsequent analysis. E shall be understood as time-independent part of the electric field.

To seek non-diffracting generalized plane wave solutions (cite paper), we assume an ansatz of the time-independent electric field E(r, θ, z) with zero longitudinal component and the transverse components dependent only on (r, θ).

E(r, θ, z) = (E 1 (r, θ)e 1 + E 2 (r, θ)e 2 ) exp[ikz] (8)
We have expressed the electric field ansatz in the Cartesian vector basis (e 1 , e 2 , e 3 ) but we will solve the Maxwell's equations in cylindrical coordinates (r, θ, z).

In cylindrical coordinates, the Laplacian operator, ∇ 2 is given by,

∇ 2 E = ∂ 2 E ∂r 2 + 1 r ∂E ∂r + 1 r 2 ∂ 2 E ∂θ 2 + ∂ 2 E ∂z 2 = ∂ 2 E ∂r 2 + 1 r ∂E ∂r + 1 r 2 ∂ 2 E ∂θ 2 -k 2 E ⇒ ∇ 2 E + k 2 E = ∂ 2 E ∂r 2 + 1 r ∂E ∂r + 1 r 2 ∂ 2 E ∂θ 2 (9)
The right hand side of the above equation can be looked as transverse part of the Laplacian operator, ∇ 2 t , while the left hand side is zero using equation ( 7). Thus, we get

∇ 2 t E = 0 or, ∇ 2 t E i = 0 ( 10
)
where i = 1, 2 correspond to the x and y components of electric field respectively. Each vector component of electric field E i satisfies Laplacian equation in polar coordinates(r, θ), whose solution is known to be of the following form [2]

E i (r, θ) = r n (a in cos(nθ) + b in sin(nθ)) (11)
where n can be any integer and a in , b in are constants to be determined from divergence free condition and boundary conditions.

Divergence free condition

∇ • E = 0 (12) ⇒ E 1,r + E 2,θ r cos θ + E 2,r - E 1,θ r sin θ = 0 (13)
Substituting E 1 (r, θ) and E 2 (r, θ) from equation (11) and simplifying using trigonometric identities, we get,

nr n-1 ((a 1n + b 2n ) cos(n -1)θ + (b 1n -a 2n ) sin(n -1)θ) = 0 (14)
Since equation ( 14) must hold true for all values of r and θ, the coefficients of each term are equated to zero.

a 1n = -b 2n b 1n = a 2n (15)
Substituting these coefficients from the above relations back into equation (11), we get,

E 1n = r n (a 1n cos(nθ) + a 2n sin(nθ)) E 2n = r n (-a 1n sin(nθ) + a 2n cos(nθ)) ( 16
)
Copyright c 2018 Saurav Aryan. ALL RIGHTS RESERVED Let R(ψ) be a rotation matrix which rotates any vector by angle ψ counter-clockwise about z-axis. Then the above solution can be written more elegantly as,

E n (r, θ, z) = r n R(-nθ) • A exp(ikz) (17)
where A = (a 1n , a 2n , 0) is a constant vector. This completes the derivation of generalized plane waves. On substituting n = 0 , we recover the "conventional" plane wave solution.

E 0 (r, θ, z) = A exp(ikz) (18)
The vector A thus, defines the polarization of the conventional vector plane wave E 0 .

Analogously, vector A is defined to be the polarization of the generalized plane wave solution for all n as well. Note that throughout our analysis, the constants a 1n , a 2n can be complex numbers, as such the notions of linearly, circularly or elliptically polarization extend to the generalized plane waves as well.

Unlike conventional vector plane waves, the generalized plane waves can exist only in certain domains depending on the sign of the charge n owing to the amplitude's dependence on r n . When n > 0, the amplitude blows up as r → ∞ and hence E n (x) for any n > 0 cannot exist in an unbounded domain. Figure [START_REF] Phillips Feynman | Mainly electromagnetism and matter[END_REF] shows the electric field given by the solution (17) with A = (1, 0, 0) and n = 1. The electric field vector rotates in the opposite sense with respect to θ, which is the case for all n > 0 and hence we term these solutions as counter -rotating generalized plane waves.

Copyright c 2018 Saurav Aryan. ALL RIGHTS RESERVED When n < 0, the amplitude blows up as r → 0 and therefore, in this case, E n (x) can only exist in a domain excluding the z-axis r = 0, such as region outside a cylinder. Figure [START_REF] Strang | Introduction to applied mathematics[END_REF] shows the electric field solution given by equation ( 17) with n = -1 and polarization

given by A = (1, 0, 0), which has radial polarization distribution. Since the electric field vector rotates in the same sense as the azimuthal angle θ when n < 0, these solutions are termed as co -rotating generalized plane waves. For the same n = -1 but A = (0, 1, 0) , azimuthally polarized electric field is produced as seen in figure [START_REF] Sarkar | General vector basis function solution of maxwells equations[END_REF]. In both figures (2) and

(3), a small region near the axis r = 0 (not visible in figures) has been excluded to keep the amplitude finite.

III. ORTHOGONALITY RELATIONS

To see that the generalized plane waves cannot be expressed as superposition of conventional plane waves, we define "inner product" of two solutions E 1 and E 2 over a region Ω

denoted by < E 1 | E 2 > Copyright c 2018 Saurav Aryan. ALL RIGHTS RESERVED < E 1 | E 2 >= Ω E 1 (x) • E * 2 (x) dx (19)
The definition is similar to that in [START_REF] Sarkar | General vector basis function solution of maxwells equations[END_REF], except that the integration is performed over the region Ω, where the field solution is valid instead of the whole region R 3 . Lets work with Ω to be a hollow cylinder 0 < a i < r < a 0 , since the generalized plane waves of any charge n can exist in bounded region not containing the axis r = 0. Then, the inner product between a generalized plane wave of charge n, E n (x) and an arbitrary conventional plane wave

F(x) = m exp(ik • x) traveling in the direction k = (k 1 , k 2 , k 3 ), | k |= k with polarization n
perpendicular to k, can be calculated as

< E n | F >= Ω dx E n (x) • F * (x) = ∞ 0 dz 2π 0 dθ ao a i rdr exp(ikz) r n R(nθ)A • m exp(-ik • x) = ∞ 0 dz exp [i(k -k 3 )z] 2π 0 dθ ao a i dr exp [ir(k 1 cos θ + k 2 sin θ)] r n-1 R(nθ)A • m (20)
The above integral vanishes for all choices of direction of propagation k of conventional plane wave F. We treat the cases, when k is along z-axis, in any other direction separately.

The integral with respect to z can be viewed as half times the Fourier transform of constant function, which is a delta function,

∞ 0 dz exp [i(k -k 3 )z] = 1 2 δ(k -k 3 ) (21)
The above vanishes for all choices of k if it is not along z-axis since k 3 = k . When k is along z-axis, k 1 = k 2 = 0, and the integral with respect to r and θ is given by

2π 0 dθ ao a i dr 1 r n-1 R(nθ)A • m = ao a i 1 r n-1 dr 2π 0 R(nθ)A • m dθ = 0 (22)
which vanishes since the integration with respect to θ of dot product of a constant vector m with R(nθ)A rotating linearly with θ is zero. More generally, any two generalized vector plane waves with different charge n traveling along the same direction, here z-axis are orthogonal, which can be proved in a similar manner as in equation ( 22). However, it is not trivial to check the orthogonality of any two generalized plane waves with different charge traveling in different direction. The expression for generalized plane wave with k not along z-axis can be written in a different cylindrical coordinate system (r , θ , z ) where z is chosen along k and r and θ are chosen appropriately so that the field solution can be given by the same expression as in (17) but with (r, θ, z) replaced by (r , θ , z ). While computing it's inner product with a field traveling along z-axis, (r , θ , z ) would need to be expressed as function of (r, θ, z). The integrand of the inner product however, can no longer be written separately in z variable as in equation ( 20), which makes it difficult to check the orthogonality.

In the above analysis, the integration with respect to r remains a finite non-zero value since it is a definite integral. However, if the domain is unbounded or contains the the axis r = 0, the integration with respect to r may no longer be finite depending on n. In particular, if the domain Ω is chosen to be the whole space R 3 , then the integration with respect to θ would still be zero, but the integration with respect to r would not be finite.

Computation of the integration of inner product (20) becomes more involved and one has to be careful in applying the above arguments to prove that the inner product still vanishes.

Further difficulty arises if the domain Ω does not possess circular symmetry and the inner product may not vanish in such cases. The restriction on n to be integer, would be relaxed in the absence of circular symmetry and additional boundary conditions would be required instead to determine possible values of n. The generalized plane waves can still exist in noncircular symmetric domain provided that the conditions for finite amplitude as discussed above are met.

IV. CONCLUSIONS

In summary, it is proved that the generalized vector plane waves cannot be expressed as superposition of conventional vector plane waves. The former constitutes a larger set of solutions to full vector Maxwell's equations, of which the later is a subset. The belief that

Copyright c 2018 Saurav Aryan. ALL RIGHTS RESERVED conventional plane wave solutions form a complete basis of solutions to vector Maxwell's equations is not true. Any other vector solutions basis, such as one presented in [START_REF] Sarkar | General vector basis function solution of maxwells equations[END_REF], shown to be complete based on this belief is not a complete basis either. It remains to be seen whether the basis formed by the generalized plane waves is complete, and also whether they are mutually orthogonal for different k and n. Further, the vector theories of diffraction which are based on the superposition of conventional plane waves could possibly be improved by including generalized vector plane waves.

How these generalized plane waves can be experimentally realized is another interesting open question. In subsequent papers, some of my theoretical attempts at designing experiment to realize simpler types of generalized plane waves will be presented. Recently, optical beams of spatially varying polarization of different kinds have gained attention and been experimentally realized. The vector solutions presented here along with other known vector solutions to Maxwell's equation [START_REF] Sarkar | General vector basis function solution of maxwells equations[END_REF][START_REF] Ww Hansen | A new type of expansion in radiation problems[END_REF][START_REF] Bouchal | Non-diffractive vector bessel beams[END_REF] should be instrumental in the development of better theoretical understanding of experimental research in vector optics.

FIG. 1 .

 1 FIG. 1. Counter-rotating generalized plane wave solution given by equation (17) with A = (1, 0, 0) and n = 1. The amplitude of the field vectors is represented by the arrow length as well as the background color (color scale in plot legend)

FIG. 2 .FIG. 3 .

 23 FIG. 2. Co-rotating generalized polarized plane wave with n = -1 and A = (1, 0, 0) has radial polarization distribution. The background color represents the magnitude of field vectors, while the arrows are of constant length indicating the direction of field.

Copyright c 2018 Saurav Aryan. ALL RIGHTS RESERVED

ACKNOWLEDGMENTS

The author would like to thank his advisers Prof. Richard James in Department of