Xinran Liang

Jeffrey E Post

Bruno Lanson

Xiaoming Wang

Mengqiang Zhu

Fan Liu

Wenfeng Tan

Xionghan Feng

Guomin Zhu

Xin Zhang

James J De Yoreo

Coupled Morphological and Structural Evolution of δ-MnO2 to α-MnO2 through multistage Oriented Assembly processes: the role of Mn(III)

Keywords: δ-MnO2, α-MnO2, morphology evolution, structure transformation, 16

published or not. The documents may come

Coupled morphological and structural evolution of δ-MnO2 to α-MnO2 through multistage oriented assembly processes: the role of Mn(III)

The δ-MnO2, as an important LMO, is associated with the α-MnO2 frequently in 88 the environmental. [START_REF] Grangeon | Cryptomelane formation from nanocrystalline vernadite precursor: a 521 high energy X-ray scattering and transmission electron microscopy perspective on 522 reaction mechanisms[END_REF] It was proposed that formation and migration of Mn(III) occurred 89 during structural transformation of δ-MnO2 into α-MnO2. [START_REF] Grangeon | Solid-state transformation of 539 nanocrystalline phyllomanganate into tectomanganate: influence of initial layer and 540 interlayer structure[END_REF] And a detailed OA process 90 by which α-MnO2 long nanorods form from α-MnO2 primary nanorods was also 91 reported. [START_REF] Yuan | Asynchronous Crystal Cell Expansion 537 during Lithiation of K + -Stabilized α-MnO2[END_REF][START_REF] Portehault | Structural and 542 morphological control of manganese oxide nanoparticles upon soft aqueous 543 precipitation through MnO4 -/Mn 2+ reaction[END_REF] However, the formation process of α-MnO2 primary nanorods from 92 poorly-crystalline δ-MnO2 nanodots is not understood. [START_REF] Portehault | Morphology Control of 545 Cryptomelane Type MnO2 Nanowires by Soft Chemistry. Growth Mechanisms in 546 Aqueous Medium[END_REF] In addition, it is unclear how 93 the new tunnel structure formation corresponds with morphology evolution by the OA 94 process. A basic knowledge of the morphology evolution across the whole synthesis 95 process and the relationship with structural transformation are also critical for the α-96 MnO2 material synthesis in application and understanding of the minerology and 97 behavior of δ-MnO2 and α-MnO2 type minerals in the nature.

98

In this study, Na + /K + -stabilized α-MnO2 nanorods were synthesized by co-99 precipitation of Na/KMnO4 with MnSO4.The dynamic process of Na + -stabilized α-100 MnO2 nanorods formation is carefully explored; because the Na phase forms at a slower 101 rate than that of K + -stabilized, as shown in previous experiments. [START_REF] Liu | Effects of Alkali Metal and 527 Ammonium Cation Templates on Nanofibrous Cryptomelane-type Manganese Oxide 528 Octahedral Molecular Sieves (OMS-2)[END_REF] , it provides a more During the synthesis of δ-MnO2, the amount of MnO4 -was in excess compared to 114 the Mn(II) shown in equation 1. Therefore, the supernatant was purple after the two 115 solutions were mixed. After the solution was heated to 100 C for 14 min, the color of 116 the supernatant changed from purple to colorless due to the complete reduction of 117 MnO4 -ions under the acidic condition (pH=0.86): [START_REF] Portehault | Structural and 542 morphological control of manganese oxide nanoparticles upon soft aqueous 543 precipitation through MnO4 -/Mn 2+ reaction[END_REF][START_REF] Mckenzie | Characterization of the 550 manganese oxide produced by pseudomonas putida strain MnB1[END_REF] , according to equation 2.

118 4MnO 4 -+4H + →4δ-MnO2 +3O 2 +2H 2 O (2)
Table 1. The AOS of the samples decrease from 3.96 to 3.70 (Table 2) before the 164 suspension was heated at 100 ℃ for 10 min. The percentages of Mn(IV) decreased from 165 83.5 % to 80.5 %, and the percentage of Mn(III) decreased from 14.3 % to 12.3 % and 166 then increased to 15.3% during this stage, indicating that the Mn(IV) and Mn(III)

167
were initially reduced to Mn(II), followed by the comproportionating between 168 vacancy-adsorbed Mn(II) and structural Mn(IV) to form Mn(III). The values of AOS 169 obtained by fitting of XPS spectra are systematically lower than that obtained by 170 titration method probably due to higher content of lower valence Mn present at the 171 surface compared to that in bulk. [START_REF] Zaharieva | Electrosynthesis, functional, and structural characterization of a water-oxidizing 575 manganese oxide[END_REF][START_REF] Yin | 577 Effects of Fe doping on the structures and properties of hexagonal birnessites -578 Comparison with Co and Ni doping[END_REF] Thus, the XPS-derived AOS represents "surface"

172

Mn AOS, and it is possible that Mn(IV) is reduced to Mn(III)/Mn(II) and adsorbed on 173 the surface of δ-MnO2 nanoparticles. [START_REF] Zaharieva | Electrosynthesis, functional, and structural characterization of a water-oxidizing 575 manganese oxide[END_REF][START_REF] Yin | 577 Effects of Fe doping on the structures and properties of hexagonal birnessites -578 Comparison with Co and Ni doping[END_REF] Despite the discrepancy between the absolute 174 transformation of layer structured Mn oxides into tunneled ones.

[-Mn(IV)-O 2--Mn(IV)-]→[-Mn(III)--Mn(III)-]+ 1 2 O 2 ( 4
)
197

The XPS fitting results of O1s are listed in S3). [START_REF] Hou | Effect of giant oxygen vacancy defects 588 on the catalytic oxidation of OMS-2 nanorods[END_REF][START_REF] Xing | An economic method to prepare vacuum 590 activated photocatalysts with high photo-activities and photosensitivities[END_REF][36][37][38] The of α-MnO2 and the percentages of Mn(III) and Mn(II) decreased gradually (Table 1).

222

And the decrease of the O1s XPS peak at 532.4 eV is induced by the reduction of 223 oxygen vacancies (Figure S3). This indicated that dissolved oxygen generated by the 224 previous reactions (equ. 3, 4) could refill into oxygen vacancies and oxidizes 225 Mn(III)/Mn(II) back to Mn(IV), causing increase of AOS from 3.70 to 3.96 (Table 2).

226

Contrarily, the content of -OH and Na + reached a maximum when δ-MnO2 started to multiple long nanosheets and primary nanorods stacking (Figure 7C).

308

The coupled morphological and structural evolution. Based on the above 309 analyses, the coupled evolutions of structure and morphology during the formation of 310 α-MnO2 can be illustrated in (Figure 8). Firstly, MnO4 -is reduced by Mn(II) to produce 311 primary nanodots of δ-MnO2 with poor crystallinity (with crystallite size of 312 approximately 2 to 4 nm). With increasing heating time, nanodots aggregate serially to 313 form long nanosheets. This process is followed by the edge-to-edge OA mechanism, by 314 which the nanodots rotate gradually to achieve the same direction within the a-b plane.

315

Secondly, the long nanosheets stack with each other to form primary nanorods, which process, the lower pH of 0.86 is favorable for edge-to-edge OA along the (100) plane. [START_REF] Marafatto | Crystal growth and aggregation in 508 suspensions of δ-MnO2 nanoparticles: implications for surface reactivity[END_REF] 326

The structure of δ-MnO2 produced in the initial crystallization stage does not meet 327 the necessary conditions (high proportion of Mn(III)/Mn(II) above/below layer for the conversion from LMOs (buserite) to TMOs (3  3 todorokite). [START_REF] Yin | Characterization of Co-doped 580 birnessites and application for removal of lead and arsenite[END_REF] The Mn(III)

333
was apparently important for the structural symmetry change and phase transformation,

334
but the specific role it plays is unclear. [44][45] The amount of Mn(IV)/(III) for the ideal 335 transition state structure can be counted as showed in Figure S7, i.e the interlayer Mn(III) 336 content requires to reach 30 mol % of total Mn to meet assembly into 2  2 α-MnO2.

337

This agrees very well with our experimental observation. The AOS of Na100-10 and conversion is much faster than that of Na + -stabilized δ-MnO2. The structure of K + -362 stabilized α-MnO2 was stable when the suspension was aged for 24 h, and previous 363 reports. [START_REF] Yuan | Asynchronous Crystal Cell Expansion 537 during Lithiation of K + -Stabilized α-MnO2[END_REF] indicate that stable (100) surface of nanorods can be formed during this stage.

364

However, when the Na + -stabilized δ-MnO2 was continuously aged under 60 ℃ for 24 365 h, the XRD patterns of the samples show a new peak dhkl = 0.4 nm of γ-MnO2 with a 1 366 × 2 tunnel structure (Figure S2). The intensity of this peak increased during aging for The measurement was performed using the rapid acquisition PDF method by

102

  photoelectron spectroscopy (XPS) to investigate the structure changes at the atomic

227 4 263

 4 Figure 5C (6.7 nm) which corresponds to the size of nanodots, which indicating the

316

  initiates the conversion of δ-MnO2 into α-MnO2. The stacking is not initially coincident 317 because of the different sizes of long nanosheets (Figure S5, 5C, D and 7C). Finally, 318 the primary nanorods assemble along the (110) and (001) planes by side-to-side and 319 end-to-end OA, respectively. 320 Although the δ-MnO2 did not convert to α-MnO2 before heating for 10 min, the δ-321 MnO2 nanodots attached with each other along their edges to form long nanosheets by 322 a dynamic network of hydrogen bonds between unsaturated oxygen either from oxygen 323 (-O), hydroxyl (-OH), or aquo-groups (-OH2) at the edges of [MnO6] octahedral layers.

  328 vacancies) for conversion to α-MnO2 . Subsequently, the structure adjustment occurs 329 through the production and migration of Mn(III) in the δ-MnO2 nanodots and long 330 nanosheets. Recently, Ling et al. showed that the amount of Mn(III) impacts the 331 structure of δ-MnO2. And Cui et al. revealed that the reduction of Mn(III) is unfavored 332

338K100- 4 ,

 4 the production of onset of transformation into 2  2 α-MnO2 is 3.70 and 3.74 339 (Mn(III) concentration is 30 % and 26 % of total Mn), respectively. Sufficient Mn(III) 340 is required for the formation the 2  2 tunnels 'walls' to support the transformation and 341 providing sufficient hydrogen bonding. One possibility is that the Mn(III) is produced 342 at low pH through the oxidation of water by layer Mn(IV) and the generation of oxygen 343 vacancies. Indeed, [Mn(III)O6] octahedra are distorted by Jahn-Teller effect, and 344 bonding distortion within the octahedral layer is relieved when Mn(III) migrates from 345 layers to the above and below the resulting vacancies (Figure 8B, step 1). 11 These 346 [Mn(III)O6] octahedra include three unsaturated oxygen which combined with H + to 347 form -OH. When the amount of -OH is large enough, a network of hydrogen bonds 348 forms between [Mn(III)O6] octahedra of adjacent nanosheets and nanodots (Figure 8B, 349 step 2, 3). These [Mn(III)O6] octahedra dehydrate via a condensation reaction and form 350 the 2  2 tunnels 'walls'. Therefore, the number of -OH is expected to be maximum at 351 the initial stage of the transformation and then to decrease during subsequent 352 transformation stages. These long nanosheets and nanodots attach along (001) plane of 353 δ-MnO2 to form α-MnO2 primary nanorods by face-to-face OA (Figure 8B, step 4). the K + -stabilized was synthesized using the similar procedures, only NaMnO4 was 356 replaced by KMnO4. During the synthesis of K + -stabilized α-MnO2, all trends of 357 structure evolution are consistent with that during Na + -stabilized α-MnO2 synthesis 358 (Figure S4 and Table S1, S2). It is suggested that the different tunnel cations α-MnO2 359 experience the similar OA formation process. The δ-MnO2 converted to α-MnO2 at 360 heating 100 ℃ for 2 min (Figure S4) indicates the rate of K + -stabilized δ-MnO2 361

367 24 h

 24 . The Na + -stabilized α-MnO2 has only small quantities of Na + (3.24 ± 0.09% in 368 Table2) in the tunnel to stabilize the tunnel structure. The α-MnO2 with empty tunnels 369 is unstable. These results indicate that the K + adsorbed on δ-MnO2 not only accelerated 370 the production and migration of Mn(III) δ-MnO2, but also promote the structural 371 conversion to and stability of the produced α-MnO2. This finding agrees well with 372 previous reports that the K + ions play important roles in templating and stabilizing the 373 tunneled framework.46 Another differences between K + -stabilized α-MnO2 and Na + -374 stabilized α-MnO2 is the dominant side crystallographic face of the products. In this 375 study, the cutting of the nanorods of Na + -stabilized α-MnO2 along the (001) plane at 376 different stages shows that the side planes of nanorods still correspond with the (110) 377 plane (Figure7D). This is perhaps in part because the open tunnel structure is formed 378 easier in the assembly process, and the morphology can be better captured during 379 slower Na synthesis. In the presence of K + , the side faces of the nanorods become four 380 (100) faces with lower energy through dissolution-recrystallize).

397Firstly,

  the poorly-crystalline δ-MnO2 nanodots aggregated along (110) surface to 398 form δ-MnO2 long nanosheets via the edge-to-edge OA mechanism. Meanwhile, the 399 Mn(IV) in the [MnO6] octahedral layer of δ-MnO2 were reduced to Mn(III) by water, 400 and then migrated to above and below the vacancies. Secondly, the morphology and 401 structure of δ-MnO2 evolved simultaneously. The Mn(III), whose amount increased 402 gradually, build up tunnel walls and triggered conversion of the 2D layer structure to 403 the 1D tunnel structure. A dynamic network of hydrogen bonds between -OH which 404 combined with Mn(III) of adjacent long nanosheets are present as a mode of bonding 405 to construct the tunnel structure. Therefore, adjacent long nanosheets aggregated and 406 formed α-MnO2 primary nanorods by a face-to-face OA mechanism. Thirdly, primary 407 nanorods assemble with each other to form longer and wider nanorods by end-to-end 408 and side-to-side OA mechanism, respectively. The defects which formed by assembly 409 are constantly smoothed via OR process throughout the whole reaction process. This 410 work fills the blank in the initial stage of 1D tunnel structured α-MnO2 formation. 411 Meanwhile, it provides greater fundamental understanding of the relationship between 412 the coupled structure and morphology transformation during crystal growth. 413 Synthesis of different types of cations of cryptomelane. In a typical experiment, 415 8.45 g MnSO4⸳H2O (0.5 M) in 100 mL of acetic acid (2 M) and 80 mL aqueous solution 416 of AMnO4 (0.4375 M, A= K or Na) were mixed together under vigorous stirring at 417 60 ℃. This suspension was heated to 100 ℃ for 20 min and was then cool to 80 ℃. 418 After cooling to 80 ℃, the suspension was frozen by liquid nitrogen and washed until 419 the conductivity of the supernatant was less than 20.0 μS cm -1 , the suspension was then 420 freeze-dried. 421 After the two solutions were mixed and the suspension was heated to 100 ℃, 10 422 mL of the suspension was extracted into a 50 mL centrifuge tube filled with 20 mL of 423 liquid nitrogen at 0, 2, 4, 6, 8, 10, 14, 16 and 20 min. The extracted samples were 424 centrifuged at 12,000 rpm for 8 min and the precipitates were washed with 50 mL of 425 DDW multiple times until the conductivity of the supernatant was less than 20.0 μS 426 cm -1 .The precipitates were then freeze-dried for characterization. Samples were 427 labelled AMix, A100-X and A80, where X is the heating time at 100 ℃ and A is the 428 type of cation. 429 Powder X-ray diffraction. The XRD patterns were collected from the dried 430 powders using a cavity mount on a Bruker D8 Advance diffractometer equipped with a 431 LynxEye detector, using Ni-filtered Cu Kα radiation (λ = 0.15418 nm). The 432 diffractometer was operated at a tube voltage of 40 kV and a current of 40 mA, with a 433 scanning rate of 1°/min and a step size of 0.02°. 434 Elemental analysis and Mn AOS. The chemical composition of the samples was 435 determined by dissolving ~ 0.1 g of each powder sample in 25 mL of 0.25 mol/L 436 NH2OH• HCl. The concentrations of dissolved Mn and K were measured using 437 inductively coupled plasma (ICP) and flame spectrometry, respectively. Mn average 438 oxidation state (AOS) was measured by a back-titration method using a KMnO4 439 standard solution. 53 440 X-ray photoelectron spectroscopy. X-ray photoelectron spectra were collected 441 using a VG Multilab2000 X-ray photoelectron spectrometer with an Al K X-ray source 442 recorded using the large area mode. The survey scans were collected using a fixed pass 444 energy of 100 eV and an energy step size of 1.0 eV, whereas the narrow scans have a 445 pass energy of 25 eV and an energy step size of 0.1 eV. The charge effect was corrected 446 by adjusting the binding energy (BE) of C (1s) to 284.62 eV. The spectra were analyzed 447 using the Avantage software. The Shirley-type background was subtracted before 448 deconvolution and fitting. The parameters used by Nesbitt et al for the multiple peaks 449 of Mn (2p3/2) for spectra fitting were adopted. 54 A 20:80 ratio of the Lorentzian: 450 Gaussian mix-sum function was used for all the fittings. 451 Transmission electron microscopy. The particle size and morphology of the 452 samples were further examined on a JEM-2100F transmission electron microscope 453 operated at an acceleration voltage of 200 kV. Prior to observation, samples were 454 embedded in epoxy resin, left to polymerize for 48 h in the dark and cut with an 455 ultramicrotome (Leica EM UC6) equipped with a diamond knife. The ~50nm thick 456 sections were picked up on lacey carbon films loaded on Cu grids. In addition to assess 457 possible preparation-induced artefacts (use of an ultramicrotome), all sample were 458 prepared according to the above described protocol, filtered and re-suspended in 459 ethanol. A drop of the obtained suspension was deposited on a copper microscope grid 460 covered with perforated carbon. 461 Field-emission scanning electron microscopy. Detailed three-dimensional 462 morphologies of particles were observed by using a field emission scanning electron 463 microscope (SU8010, Hitachi) with a maximum resolution of 1 nm. For high-resolution 464 SEM, the microscope was operated at 15 kV and the working distance was 0.5-30 mm; 465 an in-lens secondary electron detector was used. Prior to SEM analysis, each sample 466 was gold-coated.467 High-energy X-ray total scattering. Synchrotron-based X-ray total scattering 468 data were collected using an X-ray energy of 58.6491 keV (λ = 0.2114 Å) at beamline 469 11-ID-B of the Advanced Photon Source (APS), Argonne National Laboratory (APS).

  470

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. (A) XRD patterns of intermediate products at different intervals of Na + stabilized α-667 MnO2 formation and crystal growth; (B) The green zone is shown magnified on the right.. 668 669

Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .

 45678 Figure 4. TEM (A) image of Na100-6. (B) shows the jagged edges of the nanosheets. (C) particle 685 assemble on the end of nanosheets. The red circle in image C highlights the dislocation caused by 686 primary nanodots assembly. (D) The SAED pattern recorded by focusing the electron beam in the 687 area highlighted by the blue square in image A. 688 689

  [START_REF] Grangeon | Cryptomelane formation from nanocrystalline vernadite precursor: a 521 high energy X-ray scattering and transmission electron microscopy perspective on 522 reaction mechanisms[END_REF][START_REF] Yin | Characterization of Co-doped 580 birnessites and application for removal of lead and arsenite[END_REF] But the 177 mechanism by which Mn(III) is reduced from Mn(IV) in the initial t δ-MnO2 is still an MnO2, there are two possible pathways to the reduction of Mn(IV).

	178
	179	open question. As described above, some of the octahedral Mn(IV) was initially
	180	reduced to Mn(II), and a portion of that was subsequently transformed to Mn(III). Given
	181	that apart from H2O (H + ) no other reductant exists during the transformation of δ-MnO2
	to α-182
	183	One way is that the MnO2 is reduced by H + at ~ 100 ℃ and pH < 1, following the
	184	reaction: 32
		2MnO 2 +4H + +𝑒 -→2Mn 2+ +2H 2 O+O 2	(3)

185

Depending on solution pH, the Mn(II) cations originating from this partial δ-MnO2 186 dissolution might attach above/below layer vacancies.

[START_REF] Grangeon | Cryptomelane formation from nanocrystalline vernadite precursor: a 521 high energy X-ray scattering and transmission electron microscopy perspective on 522 reaction mechanisms[END_REF]

The XRD pattern of NaMix in 187

Figure. 1A shows that the crystallinity of δ-MnO2 was poor (i.e. crystallite size was 188 small), indicating large surface area, which favors equ. 3. As discussed below, however, 189 HRTEM images suggest that the crystallite sizes increased slightly during heating for 190 10 min. As the crystallinity of the sample increases, the Mn(IV) dissolution rate slows 191 down, consistent with the rate of the decrease of the AOS of reaching a minimum 192 when the suspension was heated at 100 ℃ for 10 min.

193 A second possible pathway for reduction of the Mn 4+ , is for the poorly-crystalline 194 δ-MnO2, with mixed-Mn valences (II, III and IV) to release structural oxygen, 195 following the reaction: 33 196

  Table. 1. There are three species of

	205	atoms on the opposite side of the vacancy, where there is no Mn(III), will complete
	206	their charge balance by forming OH. . Although this scenario is consistent with the
	207	observed change in XPS values for O 2-and OH -, the decrease in the XPS value for O 2-
	208	is greater relative to that of OH -than would be predicted. It is interesting that this great
	209	loss of O 2-is consistent with creation of O vacancies in the δ-MnO2, as proposed above
	210	(equation 4). Also, some studies have suggested that the observed increase of the O1s
	211	XPS peak at 532.4 eV in our XPS spectra is induced by the formation of oxygen
		vacancies (Figure
	199
	200	and oxygen in molecular water. Among the three species of oxides, the O 2-decreased
	201	from 69.9 % to 56.2 %, and the -OH increased from 18.1 % to 26.3 % during the heating
	202	of the suspension for the first 10 min. It lost three O 2-and produced six -OH When one
	203	Mn(III) migrates from [MnO6] layer to the interlayer, above or below the resulting
	204	vacancy, the Mn(III) will be coordinated to three layer O atoms and three OH; the O

198

oxygen existing in Mn oxides including lattice oxygen (O 2-), hydroxide oxygen (-OH),

  uncertainties in the XPS values and other structural

	212
	213	unknowns, however, make it impossible to draw definite conclusions. It was reported
	214	that longer bond of Mn-O, adsorbed ions, and poorly-crystallinity favored the formation
		of the oxygen vacancies.

  [START_REF] Stone | Reductive Dissolution of Manganese(III/Iv) Oxides by Substituted 586 Phenols[END_REF][39][40] Here, the increase in the content of Na + cations and

	215
	216	Mn(III) (Table1, 2), and the weakening of the crystallinity could facilitate formation of
	217	the oxygen vacancies, which promote Mn(III) production and the transformation of 2D
	218	layer structure into 1D tunnel structure.
	219	Interestingly, the intensity of O1s XPS peaks at 532.4 eV reached a maximum and
	220	AOS of δ-MnO2 reached a minimum at heating for 10 min. The reactions described in
	221	equ. 3 and equ. 4 became less favorable due to the formation of more stable structure

  [START_REF] Truong | Morphological and crystalline 530 evolution of nanostructured MnO2 and its application in lithium-air batteries[END_REF][48][49][50][51][52] The K + -stabilized α-MnO2, on the other hand, is more suitable

	387
	388	to produce long nanowires because of a stable tunnel structure with support of K + that
	389	does not collapse easily.
	390	Although this study is not exhaustive, it does highlight an important area for
	391	further work on morphology and structure interaction. Indeed, future studies could
	392	include K + ion template effects on migration of Mn(III) into tunnel 'walls' and the
		changes in reactivities of the products which are formed in different stages.

393

CONCLUSION 394 Na + -stabilized tunnel structured α-MnO2 nanorods are found to assemble via the 395 staged oriented attachment (OA) growth process. The coupling evolution of structures 396 and morphologies during the whole process has been explored.

Table 2 .

 2 Perkin Elmer amorphous silicon detector. The image plate was exposed 472 for 1.2 s and the measurement was repeated 75 times for a total collection time of 90 s 473 for each sample. The software Fit2D was used to integrate and convert the 2-D raw data Xiao, W.; Xia, H.; Fuh, J. Y. H.; Lu, L., Growth of single-crystal α-MnO2 nanotubes 640 prepared by a hydrothermal route and their electrochemical properties. J. Power. Na + content, average oxidation states (AOS) of Mn in intermediate products at various 662 time intervals during Na + -stabilized α-MnO2 formation obtained from titration and fittings of Mn 663

	474
	475	to 1-D intensity versus wave vector (Q) data. The PDF, G(r), was obtained from the
		raw 1-D data using the program PDFgetX2.
	477
	478	We thank Dr. GV Lowry (AE) and the anonymous reviewers for thoughtful
	479	reviews and constructive comments. The authors thank the National Natural Science
	480	Foundation of China (Grant Nos. 41471194 & 41171197) and the Strategic Priority
	481	Research Program of the Chinese Academy of Sciences (No. XDB15020402) for
	482	financial support of this research. Use of the Advanced Photon Source, Argonne
	483	National Laboratory, supported by U.S. DOE-BES under Contract DE-AC02-
	484	06CH11357.
	485
	486
	487
	488
	489
	490

471

employing a 639 52.

Acknowledgements


Nicolas C M Marty
email: n.marty@brgm.frhighlights

Sylvain Grangeon

Arnault Lassin

Benoit Madé

Philippe Blanc

Bruno Lanson

A quantitative and mechanistic model for the coupling between chemistry and clay hydration

Keywords: Clay composition, ionic strength, hydration model, exchange reaction, Gapon, Rothmund-Kornfeld

 smectite water vapor isotherms modelled using an ion exchange formalism  model accounting for the interlayer composition and relative humidity  model accounting for the charge location  hydration and stability of the clay layer processed independently

Introduction

The crystal structure of clay minerals can be described as the parallel stacking of layers separated from each other by an interlayer space. Smectite, a generic name that encompasses several minerals [START_REF] Guggenheim | Summary of recommendations of nomenclature committees relevant to clay mineralogy: Report of the Association Internationale pour l'Etude des Argiles (AIPEA) Nomenclature[END_REF], is one of the most abundant type of clay minerals in various natural settings, including soils and sediments [START_REF] Griffin | The distribution of clay minerals in the World Ocean[END_REF][START_REF] Jackson | Frequency Distribution of Clay Minerals in Major Great Soil Groups as Related to the Factors of Soil Formation[END_REF][START_REF] Murray | Effect of Weathering on Clay Minerals[END_REF], together with illite and chlorite. The general layer structure of smectite can be described as two tetrahedral sheets sandwiching an octahedral sheet. In the octahedral sheet, cations can be mainly divalent (trioctahedral smectite) or trivalent (dioctahedral smectite). The isomorphic substitution of a cation by another of lower valence (e.g. Al 3+ for Si 4+ in the tetrahedral sheet, Mg 2+ for Al 3+ or Li + for Mg 2+ in the octahedral sheet)

gives rise to a permanent layer charge that is compensated for by the presence of hydrated cations in the interlayer space. Vermiculites have a similar layer structure and differ from smectites by their higher layer charge deficit [1.2-1.8 and 0.4-1.2 charge equivalent per O 20 (OH) 4 in vermiculites and smectites, respectively - [START_REF] Guggenheim | Summary of recommendations of nomenclature committees relevant to clay mineralogy: Report of the Association Internationale pour l'Etude des Argiles (AIPEA) Nomenclature[END_REF]]. This charge contrast does not appear to induce significant modification of the hydration behavior or of the distribution of interlayer species [START_REF] Dazas | Influence of Tetrahedral Layer Charge on the Organization of Interlayer Water and Ions in Synthetic Na-Saturated Smectites[END_REF][START_REF] Vinci | Influence of layer charge on hydration properties of synthetic octahedrally-charged Nasaturated trioctahedral swelling phyllosilicates[END_REF]. Both families of swelling phyllosilicates are thus hereafter jointly referred to as "smectite". The classification of smectite minerals is based on multiple criteria and takes into account the tri-or dioctahedral character of the octahedral sheets, the location of the layer charge (tetrahedral or octahedral), and the density of the layer charge [START_REF] Guggenheim | Summary of recommendations of nomenclature committees relevant to clay mineralogy: Report of the Association Internationale pour l'Etude des Argiles (AIPEA) Nomenclature[END_REF]. For example, saponite is a trioctahedral smectite with tetrahedral layer charge, hectorite is a trioctahedral smectite with octahedral layer charge, and montmorillonite is a dioctahedral smectite with octahedral layer charge. The density and location of the layer charge influences not only smectite cation uptake capacity and the relative affinity of different cations for its surface, but also smectite hydration properties [START_REF] Sato | Effects of Layer Charge, Charge Location, and Energy Change on Expansion Properties of Dioctahedral Smectites[END_REF][START_REF] Vinci | Influence of layer charge on hydration properties of synthetic octahedrally-charged Nasaturated trioctahedral swelling phyllosilicates[END_REF]. In turn, hydration and dehydration properties influence smectite swelling and shrinkage properties (e.g. [START_REF] Norrish | The swelling of montmorillonite[END_REF]. Indeed, smectite hydration involves the uptake of interlayer water, i.e. the intercalation of a variable number of water molecules in the initially anhydrous interlayer (hereafter referred to as the "0W" state). These water molecules are organized to form 1, 2, or 3 "planes" parallel to the layer plane (hereafter denominated "1W", "2W", and "3W", respectively), with the most hydrated interlayer having the highest number of "water planes" and the highest layer-to-layer distance. These interlayer hydration states often coexist within a given crystal, and the structure is then described as "interstratified", with distinct interlayer spaces hosting different number of water planes. The quantitative understanding of clay hydration thus requires not only the modelling of the total amount of water sorbed as a function of the relative humidity, but also the quantification of the relative abundances of the different hydration states. This ability to model clay hydration is a fundamental stage in the development of THMC (Thermal-Hydraulic-Mechanical-Chemical) codes, which aim to couple chemistry, thermal, hydraulic and mechanical effects to be as realistic as possible, and thus require the link between layer charge and hydration to be quantitatively assessed.

To describe the water desorption/adsorption isotherms on clays, numerous studies used the solid solution model based on [START_REF] Ransom | A chemical and thermodynamic model of aluminous dioctahedral 2:1 layer clay minerals in diagenetic processes; regular solution representation of interlayer dehydration in smectite[END_REF]Helgeson (1994) (e.g. Dubacq et al., 2009;[START_REF] Gailhanou | Methodology for determining the thermodynamic properties of smectite hydration[END_REF][START_REF] Tajeddine | Hydration-dehydration behavior and thermodynamics of MX-80 montmorillonite studied using thermal analysis[END_REF][START_REF] Vidal | Thermodynamic modelling of clay dehydration, stability and compositional evolution with temperature, pressure and H2O activity[END_REF][START_REF] Vieillard | Hydration thermodynamics of the SWy-1 montmorillonite saturated with alkali and alkaline-earth cations: A predictive model[END_REF][START_REF] Vieillard | A predictive model of thermodynamic entities of hydration for smectites: Application to the formation properties of smectites[END_REF]. Although solid solutions, or the consideration of discretized clay end-members with different hydration states, can be employed to describe the water content of clays, the practical implementation of this approach in reactive transport codes remains difficult. These models require indeed the thermodynamic properties of each end-member to be known, i.e. each hydration state, for each layer charge and each interlayer cation, has to be characterized together with possible interaction parameters between end-members (e.g. Margules parameters). Furthermore, most of these models focus on clay formation or stability, and therefore include thermodynamic properties of clay layers (i.e. standard Gibbs energy of formation). As changes in hydration state are relatively rapid processes (e.g. [START_REF] Bray | The kinetics of dehydration in Camontmorillonite; an in situ X-ray diffraction study[END_REF], solid solutions that relate the stability of clay layers are thus incompatible with the use of kinetics driving slow dissolution/precipitation rates of clay minerals (e.g. [START_REF] Marty | Weathering of an argillaceous rock in the presence of atmospheric conditions: A flow-through experiment and modelling study[END_REF].

Another drawback of the solid solution approach when applied to geochemical modelling is the requirement for the cross-linked thermodynamic equilibrium of both the clay layer and the interlayer cation composition with the contacting solution. As an alternative to these thermodynamic models, Freundlich's model and the BET model (Brunauer, Emmett and Teller) (e.g. [START_REF] Hatch | Water Adsorption on Clay Minerals As a Function of Relative Humidity: Application of BET and Freundlich Adsorption Models[END_REF][START_REF] Revil | Unified water isotherms for clayey porous materials[END_REF], as well as other equations described in [START_REF] Arthur | Evaluation of theoretical and empirical water vapor sorption isotherm models for soils[END_REF], are commonly used to reproduce clay desorption/adsorption isotherms. Despite significant efforts (e.g. [START_REF] Lindholm | Deconvolution of Smectite Hydration Isotherms[END_REF], the lack of mechanistic constraints relating the chemical properties of clay makes the Mechanical-Chemical coupling uncertain, however.

Most limitations of the above-described modelling approaches can be circumvented by molecular modelling techniques, which are increasingly used to study the conformation of atoms in the structure (e.g. [START_REF] Dazas | Influence of Tetrahedral Layer Charge on the Organization of Interlayer Water and Ions in Synthetic Na-Saturated Smectites[END_REF][START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 2. Toward a Precise Coupling between Molecular Simulations and Diffraction Data[END_REF], including the dynamics of interlayer cations and water (e.g. [START_REF] Holmboe | Molecular Dynamics Simulations of Water and Sodium Diffusion in Smectite Interlayer Nanopores as a Function of Pore Size and Temperature[END_REF][START_REF] Rotenberg | Modelling water and ion diffusion in clays: A multiscale approach[END_REF].

Despite its numerous advantages (sound mechanistic foundations, capacity to probe the kinetics of ion exchange, capacity to predict), the large computational expense required makes molecular modelling incompatible with large scale reactive transport simulations.

The present study aims to develop an alternative model focused on clay reactivity (i.e. hydration and cation exchange) which is both numerically robust and easy to implement in geochemical codes, for example to reproduce diagenetic processes (e.g. [START_REF] Tremosa | Effects of smectite dehydration and illitisation on overpressures in sedimentary basins: A coupled chemical and thermo-hydromechanical modelling approach[END_REF], soil shrink-swell (e.g. [START_REF] Cornelis | Measuring and modelling the soil shrinkage characteristic curve[END_REF], soil water availability (e.g. [START_REF] Rawls | Estimating Soil Water Retention from Soil Physical Properties and Characteristics[END_REF] or the fate of clays at a radioactive waste disposal site (e.g. [START_REF] Marty | Simulation of Cement/Clay Interactions: Feedback on the Increasing Complexity of Modelling Strategies[END_REF]. The proposed model is based on the Gapon convention [START_REF] Gapon | On the theory of exchange adsorption in soils[END_REF] and uses the exchange reaction formalism (i.e. approach similar to an ideal solid solution) to quantitatively reproduce water desorption isotherms, including total water content and clay hydration heterogeneity (i.e. proportions of 0W, 1W, 2W, and 3W hydrates within a given crystal). It also takes into account the density and location of the layer charge deficit (octahedral or tetrahedral) and the nature of the interlayer cation. As it is based on the exchange convention, this model makes it possible to reproduce quantitatively cation exchange reactions. It was validated against a variety of experimental datasets, including desorption isotherms of two saponites and two hectorites with contrasting layer charges and of Wyoming montmorillonite (SWy-1) with various interlayer compositions (Na and Ca). The present model is satisfactory from a phenomenological point of view as water is presumed to have a specific affinity for each cation and is not considered as an independent parameter, as in many alternative models. Finally, the hydrated exchange models were compared to the classic exchange approach, which conventionally involves anhydrous reactions, and data extracted from experiments involving clay in contact with saline solutions.

Materials and methods

Background

A hydrated smectite powder has three different water "reservoirs" (e.g. see figure 1 in [START_REF] Gailhanou | Methodology for determining the thermodynamic properties of smectite hydration[END_REF]. The main one is the interlayer space, where water molecules are hydrating interlayer cations to different degrees (i.e. hydration sphere) according to layer charge, nature of interlayer cation, temperature, and relative humidity (e.g. [START_REF] Dazas | Influence of Tetrahedral Layer Charge on the Organization of Interlayer Water and Ions in Synthetic Na-Saturated Smectites[END_REF].

The other two reservoirs are the "external water", distributed across the external surfaces of clay tactoids (composed of a variable number of stacked layers) [START_REF] Salles | Hydration of swelling clays: multi-scale sequence of hydration and determination of macroscopic energies from microscopic properties[END_REF], and the pore water, located between the clay aggregates (built of connected clay tactoids). Pore water was not accounted for in the present study owing to its negligible role in clay swelling. Moreover, the distinction between interlayer and external water was not made in the present study. Indeed, only interlayer water was required to reproduce water isotherms up to a relative humidity of ~60-70 %, in agreement with previous X-ray diffraction, neutron diffraction, and molecular modelling studies (Ferrage et al., 2005a[START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF][START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 2. Toward a Precise Coupling between Molecular Simulations and Diffraction Data[END_REF][START_REF] Vinci | Influence of layer charge on hydration properties of synthetic octahedrally-charged Nasaturated trioctahedral swelling phyllosilicates[END_REF]. A low contribution of external water in such conditions was also supported by [START_REF] Lindholm | Deconvolution of Smectite Hydration Isotherms[END_REF].

Its contribution is only significant in the relative humidity range where pore water is also present.

Hydrated exchange reactions

The geochemical selectivity constant that relates the amount of a given species in solution with those adsorbed on the solid depends on the water content of the studied material (e.g. [START_REF] Redinha | Thermodynamics of ion-exchange processes. System H, Na and Ag with polystyrene sulphonate resins[END_REF][START_REF] Steck | Water sorption and cation-exchange selectivity of a perfluorosulfonate ion-exchange polymer[END_REF]. Regarding clay minerals, [START_REF] Tardy | A method of estimateng the Gibbs free energies of formation of hydrated and dehydrated clay minerals[END_REF] described a dependency of the Na/Ca exchange reaction on the number of water molecules. Similarly, [START_REF] Whittaker | Ion exchange selectivity in clay is controlled by nanoscale chemical-mechanical coupling[END_REF] have proposed a Na/K exchange model in which the number of water planes was correlated to water activity. To take into account this effect, the Gapon convention [START_REF] Gapon | On the theory of exchange adsorption in soils[END_REF] was used here to model the ion exchange reactions involving water molecules in the interlayer space. This convention expresses exchange reaction per mol of exchange sites rather than per mol of exchanging ion [START_REF] Gaines | Adsorption studies on clay minerals. II. A formulation of the thermodynamics of exchange adsorption[END_REF]. For monovalent Na + and bivalent Ca 2+ cations, the reaction is thus written in the following way:

X𝑁𝑎 + 0.5𝐶𝑎 2+ ↔ 𝑁𝑎 + + X𝐶𝑎 0.5 (Eq. 1)
where X -represents the exchanger to which Na + or Ca 2+ cation is bound.

The distribution of species is given by the law of mass action:

𝐾 𝑁𝑎/𝐶𝑎 = 𝐸 𝑋𝐶𝑎 0.5 𝐸 𝑋𝑁𝑎 × [𝑁𝑎 + ] [𝐶𝑎 2+ ] 0.5 (Eq. 2)
where 𝐾 𝑁𝑎/𝐶𝑎 is the selectivity constant of the reaction (Eq. 1), E referring to the equivalent fraction of species on the exchanger and brackets referring to the activity of Na + or Ca 2+ in solution.

In the following, hydration reactions were defined in reference to the dry state. Hydration reactions involving Na + then write:

𝑋𝑁𝑎 + 𝑛𝐻 2 𝑂 ↔ 𝑋𝑁𝑎(𝐻 2 𝑂) 𝑛 (Eq. 3)
where n is the number of water molecules in the interlayer space per layer charge.

For bivalent cations such as Ca 2+ , the hydration reactions are written as follows:

𝑋𝐶𝑎 0.5 + 𝑛𝐻 2 𝑂 ↔ X𝐶𝑎 0.5 (𝐻 2 𝑂) 𝑛 (Eq. 4)
Following the Gapon convention, the general expression of mass action laws for hydration reactions is:

𝐾 𝑊 = 𝐸 𝑋𝐶𝑎𝑡𝑖𝑜𝑛(𝐻 2 𝑂) 𝑛 𝐸 𝑋𝐶𝑎𝑡𝑖𝑜𝑛 × 1 [𝐻 2 𝑂] 𝑛 (Eq. 5)
where K w is the thermodynamic constant of the hydration reaction (Eq. 3 or 4) and XCation refers to XNa (Eq. 3) or XCa 0.5 (Eq. 4).

The definition of Eq. 3-5 illustrates interactions between water, the clay layer and interlayer cation. If necessary, the previous equation (Eq. 5) can be modified according to the Rothmund-Kornfeld description [START_REF] Bond | On the Rothmund-Kornfeld Description of Cation Exchange[END_REF]:

𝐾 𝑊 = 𝐸 𝑋𝐶𝑎𝑡𝑖𝑜𝑛(𝐻 2 𝑂) 𝑛 𝐸 𝑋𝐶𝑎𝑡𝑖𝑜𝑛 × ( 1 [𝐻 2 𝑂] ) 𝛽 (Eq. 6)
where β is an empirical parameter. Note that Gapon and Rothmund-Kornfeld approaches are equivalent for β = n.

Modelling strategy

The calculations were performed with PHREEQC3 [START_REF] Parkhurst | Description of input and examples for PHREEQC version 3-a computer program for speciation, batch-reaction, one-dimensional transport, and inverse geochemical calculations[END_REF] and the thermodynamic database THERMOCHIMIE version 9a [START_REF] Blanc | ThermoChimie database developments in the framework of cement/clay interactions[END_REF][START_REF] Giffaut | Andra thermodynamic database for performance assessment: ThermoChimie[END_REF].

The simulations of water vapor sorption isotherms required the definition of pore water since the involved chemical reactions do not occur nor can be described in the absence of solvent.

The cation concentration in pore water was set at 1 meq L -1 (i.e. solutions containing NaCl, CaCl 2 or NaCl/CaCl 2 , with Cl concentration = 1 mmol L -1 ). The salinity of the pore water was then adjusted by imposing a partial pressure of H 2 O(g) to control water activity. In addition, for a solution containing both Na + and Ca 2+ aqueous species, the increase in the ionic strength of the solution modifies the cationic composition of the exchanger (e.g. [START_REF] Appelo | Geochemistry, groundwater and pollution[END_REF][START_REF] Fletcher | Chemical modeling of clay/electrolyte interactions of montmorillonite[END_REF]. To overcome this effect and reach conditions expected for water adsorption/desorption experiments, an extremely low quantity of pore water can be set in the system (i.e. < 0.1 g H 2 O for 1 g of clay). In doing so, the quantity of cations present outside the smectite is much smaller than inside the interlayer spaces and the effect of salinity on the cationic composition of the exchanger can thus be disregarded. The initial Na + /Ca 2+ interlayer population is then independent of relative humidity, as expected during measurements of water vapor sorption isotherms. Furthermore, as the water activity is set in the models [set partial pressure of H 2 O(g)] and the effect of salinity on exchangeable cations can be disregarded, simulations of water vapor isotherms do not require the use of activity coefficient corrections dedicated to high salinities, such as the Pitzer equations and their associated databases (e.g. [START_REF] Lach | A Pitzer Parametrization To Predict Solution Properties and Salt Solubility in the H-Na-K-Ca-Mg-NO 3 -H 2 O System at 298.15[END_REF][START_REF] Lassin | Solution properties and salt-solution equilibria in the H-Li-Na-K-Ca-Mg-Cl-H2O system at 25 °C: A new thermodynamic model based on Pitzer's equations[END_REF].

Data selection

The water contents obtained along the adsorption branch of the isotherms are systematically lower than those measured during desorption (e.g. [START_REF] Bérend | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonites; 2, The Li + , Na + , K + , Rb + and Cs + -exchanged forms[END_REF][START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite. 1. The sodium-exchanged form[END_REF][START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF][START_REF] Lindholm | Deconvolution of Smectite Hydration Isotherms[END_REF], as the result of various and complex processes (e.g. [START_REF] Woodruff | CEC-normalized clay-water sorption isotherm[END_REF]) that disfavor water uptake. In the macroscopic approach proposed here, this could be addressed by using an energetic barrier corresponding to the energy required to re-expand the layer-to-layer distance. In addition, clay hydration along the desorption branch of the isotherms appears to be more homogeneous compared to the sorption branch (e.g. [START_REF] Bérend | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonites; 2, The Li + , Na + , K + , Rb + and Cs + -exchanged forms[END_REF]. Consequently, simulations focused on desorption branch of the isotherms for which no energetic barrier is involved.

Water desorption data of three different smectite minerals (saponite, hectorite, and montmorillonite) were taken from the literature. They were selected based on (i) the purity of investigated samples, (ii) data quality, and (iii) availability of results from complementary methods (e.g. X-ray diffraction) that allowed quantifying the relative proportion of the different hydration states. For saponite, data from [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF] were used. This study provides data for two samples containing interlayer Na and differing in their tetrahedral charge [0.8 and 1.4 charge equivalent per O 20 (OH) 4 ]. These two samples are hereafter referred to as S-Na 0.8 and S-Na 1.4 , respectively. For hectorite, we used data from [START_REF] Vinci | Influence of layer charge on hydration properties of synthetic octahedrally-charged Nasaturated trioctahedral swelling phyllosilicates[END_REF]. These authors studied the hydration of several sodium hectorites having octahedral charges ranging from 0.8 (hereinafter referred to as H-Na 0.8 ) to 1.6 (hereinafter referred to as H-Na 1.6 ) equivalent per O 20 (OH) 4 . Finally, for montmorillonite, the Wyoming montmorillonite (SWy-1) with a charge deficit of 0.74 equivalent per O 20 (OH) 4 (e.g. [START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite. 1. The sodium-exchanged form[END_REF], mainly located in the octahedral sheet (e.g. [START_REF] Sato | Effects of Layer Charge, Charge Location, and Energy Change on Expansion Properties of Dioctahedral Smectites[END_REF], was selected. The SWy-1 was used to study the effect of the interlayer cation nature (Na, Ca, and mixed Na/Ca) on water desorption isotherms. For this natural sample, changes of hydration state (e.g. from 2W to 1W) are less marked than those observed with synthetic materials listed above, possibly as the result of an increased surface charge heterogeneity compared to synthetic saponites and hectorites. Up to 3 water layers (3W) have been identified at highest relative humidities for hectorite [START_REF] Vinci | Influence of layer charge on hydration properties of synthetic octahedrally-charged Nasaturated trioctahedral swelling phyllosilicates[END_REF] and montmorillonite [START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite. 1. The sodium-exchanged form[END_REF][START_REF] Holmboe | Porosity investigation of compacted bentonite using XRD profile modeling[END_REF]. [START_REF] Dazas | Interlayer structure model of trihydrated low-charge smectite by X-ray diffraction and Monte Carlo modeling in the Grand Canonical ensemble[END_REF] showed that the 3W layer cannot be described as 3 perfectly discrete planes, but that part ot its interlayer water has some characteristics of the bulk water. Overall, we assumed that the 3W layer is involved in the crystalline swelling rather the osmotic swelling (e.g. [START_REF] Madsen | The swelling behaviour of clays[END_REF][START_REF] Norrish | The swelling of montmorillonite[END_REF] and therefore, we considered it in our modelling exercise.

Results

Effect of smectite tetrahedral layer charge (saponite)

The total amount of water incorporated in S-Na 0.8 as a function of relative humidity (Figure 1a)

and the corresponding relative proportions of the different hydration states (Figure 1b), extracted from [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF], show that the total interlayer water content is correlated to the variation in the proportions of 0W, 1W and 2W interlayer hydration states.

Table 1. Reactions and selectivity constants used to simulate the dehydration of saponites S-Na0.8 and S-Na1.4. For 1 g of S-Na0.8, a charge deficit of 0.8 eq mol -1 and a molar mass of 776 g mol -1 [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF], the exchanger quantity is 10 -3 mol. For 1 g of S-Na1.4, a charge deficit of 1.4 eq mol -1 and a molar mass of 789 g mol -1 [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF], the exchanger quantity is 1.8 10 -3 mol. S-Na 0.8 data were reproduced using the reactions and selectivity constants listed in Table 1. It was assumed that the species corresponding to 0 water layers did not involve water molecules (XNa in Table 1), although water molecules bound to the interlayer cation likely persist even at the lowest relative humidity (e.g. [START_REF] Bérend | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonites; 2, The Li + , Na + , K + , Rb + and Cs + -exchanged forms[END_REF][START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite. 1. The sodium-exchanged form[END_REF][START_REF] Kloprogge | The interlayer collapse during dehydration of synthetic Na0. 7-beidellite: a 23Na solid-state magic-angle spinning NMR study[END_REF][START_REF] Rinnert | Hydration of a Synthetic Clay with Tetrahedral Charges: A Multidisciplinary Experimental and Numerical Study[END_REF].

Clay

The quantity of water in the 1W (Table 1) was estimated to 5 10 -3 mol H 2 O g -1 of clay, using the Figure 7 in [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF]. Interestingly, this value is independent of the cation exchange capacity (CEC), an observation supported by [START_REF] Dazas | Influence of Tetrahedral Layer Charge on the Organization of Interlayer Water and Ions in Synthetic Na-Saturated Smectites[END_REF]. Consequently, the amount of water to be considered in hydration reaction (Eq. 3) was corrected by the total exchangeable quantity following:

𝑛 1𝑊 = 5 10 -3 ×m 𝑐𝑙𝑎𝑦 𝑛 𝑋𝑁𝑎 (Eq. 7)
where m 𝑐𝑙𝑎𝑦 is the clay mass considered in the numerical simulation and 𝑛 𝑋𝑁𝑎 is the corresponding exchanger quantity (in moles).

For 1 g of clay, a charge deficit of 0.8 equivalent per mole of clay and a molar mass of 776 g mol -1 [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF], the exchanger quantity is 10 -3 mol and one obtain 𝑛 1𝑊 = 5.

Likewise, for 2 water layers (2W in Table 1), we found:

𝑛 2𝑊 = 12 10 -3 ×m 𝑐𝑙𝑎𝑦 𝑛 𝑋𝑁𝑎 = 12 (Eq. 8)
Overall, amounts of water calculated for 1W and 2W reactions agree with several works, including molecular modelling techniques (e.g. [START_REF] Dazas | Interlayer structure model of trihydrated low-charge smectite by X-ray diffraction and Monte Carlo modeling in the Grand Canonical ensemble[END_REF][START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 2. Toward a Precise Coupling between Molecular Simulations and Diffraction Data[END_REF][START_REF] Holmboe | Molecular Dynamics Simulations of Water and Sodium Diffusion in Smectite Interlayer Nanopores as a Function of Pore Size and Temperature[END_REF]. Both the desorption isotherm of [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF] and the relative contribution of the different types of layers were satisfyingly modelled (Figure 1) with these parameters and refined selectivity constants (Table 1 and electronic annex S-Na0.8.pqi).

The misfit at the highest relative humidity (grey area in Figure 1a) in the desorption isotherm was due to the presence of pore water [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF][START_REF] Salles | Driving force for the hydration of the swelling clays: Case of montmorillonites saturated with alkaline-earth cations[END_REF] which was not considered in this study. Parameters and selectivity coefficients describing the hydration reaction of S-Na 1.4 were determined as for S-Na 0.8 and are shown in Table 1. Selectivity constants required further adjustment owing to the change in the number of water molecules involved in the hydration reactions (Eq. 3). As for S-Na 0.8 , both the desorption isotherms (Figure 2a) and the relative contribution of the different types of layers (0W, 1W and 2W -Figure 2b) as a function of relative humidity quantitatively reproduced the experimental data of [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 1. Results from X-ray Diffraction Profile Modeling[END_REF]. Our modelling accounted also for the observed broadening of the transition zone between 1W and 2W hydration states, which was due to a decrease of the exponents involved in mass action equations (2.8 and 6.7 for 1W and 2W reactions, respectivelysee Eq. 5 and Table 1).

Together with the data from S-Na 0.8 , this successful data modelling showed that Gapon approach can model and predict Na-saponite hydration. et al. (2010). Curves: numerical results. Same color coding as in Figure 1.

Discussion

A dependency on charge location was implemented here in the formalism used here to describe and model hydration reactions. The Gapon formalism was able to predict saponite hydration (S-Na 0.8 and S-Na 1.4 ), for which the clay layer and interlayer Na + , and associated H 2 O molecules, are strongly bound owing to the strong undersaturation of layer surface oxygen atoms (i.e. tetrahedral charge, [START_REF] Michot | Hydration and swelling of synthetic Na-saponites: Influence of layer charge[END_REF][START_REF] Prost | State and location of water adsorbed on clay minerals; consequences of the hydration and swelling-shrinkage phenomena[END_REF][START_REF] Skipper | The structure of interlayer water in a hydrated 2:1 clay[END_REF].

Further investigations are required however to assess the validity of this model for other cations (e.g. calcium saponite). The Rothmund-Kornfeld description was used for the modelling of hectorite and montmorillonite hydration (H-Na 0.8 , H-Na 1.6 , Na-SWy-1 and Ca-SWy-1), for which water molecules interact weakly with the clay layer (i.e. octahedral charge, [START_REF] Doner | Charge Location as a Factor in the Dehydration of 2:1 Clay Minerals[END_REF][START_REF] Prost | State and location of water adsorbed on clay minerals; consequences of the hydration and swelling-shrinkage phenomena[END_REF][START_REF] Vinci | Influence of layer charge on hydration properties of synthetic octahedrally-charged Nasaturated trioctahedral swelling phyllosilicates[END_REF]. The later empirical description may also be used to describe saponite hydration, however, as both Gapon and Rothmund-Kornfeld approaches are equivalent when β = n (Eq. 5-6). Regarding ion exchange reactions, the Rothmund-Kornfeld description has been found to be applicable in numerous cases (e.g. [START_REF] Bond | Comparison of Methods for Predicting Ternary Exchange from Binary Isotherms[END_REF][START_REF] Escudey | Gaines-thomas and rothmund-kornfeld descriptions of potassium-calcium exchange on variable surface charge soils[END_REF][START_REF] Reynolds | A Rothmund-Kornfeld model of Cs + -K + -Na + exchange on spherical resorcinol-formaldehyde (sRF) resin in Hanford nuclear waste[END_REF]. As reported by [START_REF] Bond | On the Rothmund-Kornfeld Description of Cation Exchange[END_REF], it appears more accurate because it avoids extrapolation of the data required for the thermodynamic approach.

The capacity of our model to predict the complex hydration behavior of clays exposed to contrasting environmental conditions was evaluated for a smectite interacting with a mixed cation pore water composition using parameters previously established for Na-SWy-1 and Ca-SWy-1 (Table 3). The desorption isotherm calculated for the clay SWy-1 containing ~50% of Na + and ~25% of Ca 2+ is shown in Figure 6a (green curve). The PHREEQC input file is given in Electronic Annex (see SWy-1_NaCa.pqi). This isotherm was calculated as a simple weighted sum of water layer contributions from Na and Ca end-members. Such an assumption relies on the presence of homoionic interlayers and is supported by numerous works showing a segregation of cations in different interlayers (e.g. Ferrage et al., 2005c;[START_REF] Méring | Sur le rôle de la valence des cations échangeables dans la montmorillonite[END_REF][START_REF] Möller | Facile Scalable Synthesis of Rectorites[END_REF]. Nonetheless, cation de-mixing is a complex phenomenon that depends on several parameters such as the substitution level [START_REF] Fink | Demixing of Exchangeable Cations in Free-Swelling Bentonite Clay[END_REF] or the layer charge [START_REF] Laird | Influence of layer charge on swelling of smectites[END_REF]. The desorption isotherm of the smectite Na/Ca-SWy-1 was then found in between that of the smectite Na-SWy-1 (red curve) and that of Ca-SWy-1 (blue curve). A similar behavior has been observed by [START_REF] Keren | Water Vapor Isotherms and Heat of Immersion of Na/Ca-Montmorillonite Systems-II: Mixed Systems[END_REF] on the adsorption isotherm of a mixed Na/Ca Wyoming. Consistent with the adopted simulation approach (see section 1.3), and as expected during an isotherm measurement, the proportion of Na and Ca in the interlayer space was independent of relative humidity (Figure 6b). The proposed model allowed also predicting the effect of solution salinity on the interlayer composition of the SWy-1 (Figure 7). In accordance with the cation exchange data of [START_REF] Gaucher | A robust model for pore-water chemistry of clayrock[END_REF], the selectivity constant of the anhydrous Na/Ca exchange reaction (Eq. 1) was equal to 0.35 (log unit). At lowest salinities, the presence of most hydrated species (i.e. 3W

hydration reactions reported in Table 3) were expected from previous considerations, however, and thermodynamic exchange constant must be corrected when Eq. 1 is coupled with Eq. 3-4:

𝐿𝑜𝑔 𝐾 𝑁𝑎/𝐶𝑎 ℎ𝑦𝑑𝑟𝑎𝑡𝑒𝑑 = 𝐿𝑜𝑔 𝐾 3𝑊 𝑁𝑎-𝑆𝑊𝑦-1 -𝐿𝑜𝑔 𝐾 3𝑊 𝐶𝑎-𝑆𝑊𝑦-1 + 𝐿𝑜𝑔 𝐾 𝑁𝑎/𝐶𝑎 𝑎𝑛ℎ𝑦𝑑𝑟𝑜𝑢𝑠 (Eq. 9)

where 𝐾 𝑁𝑎/𝐶𝑎 ℎ𝑦𝑑𝑟𝑎𝑡𝑒𝑑 is the selectivity constant for hydrated Na/Ca exchange reaction, 𝐾 3𝑊 𝑁𝑎-𝑆𝑊𝑦-1

the thermodynamic constant of the 3W hydration reaction involving the Na-SWy-1 (Table 3), 𝐾 3𝑊 𝐶𝑎-𝑆𝑊𝑦-1 the thermodynamic constant of the 3W hydration reaction involving the Ca-SWy-1 (Table 3) and 𝐾 𝑁𝑎/𝐶𝑎 𝑎𝑛ℎ𝑦𝑑𝑟𝑜𝑢𝑠 the selectivity constant of the anhydrous Na/Ca exchange reaction (10 0.35 ).

The Na/Ca exchange isotherm has been calculated for total Na and Ca concentrations of 0.01 and 1 mol L -1 (Cl -as charge-compensating anion). As showed in numerous studies (e.g. [START_REF] Appelo | Geochemistry, groundwater and pollution[END_REF][START_REF] Fletcher | Chemical modeling of clay/electrolyte interactions of montmorillonite[END_REF], calculations depend on the exponent used in the mass action equation (Eq. 2) and thus the content of adsorbed Ca 2+ increases with the decrease of the ionic strength. The exchange isotherms obtained with the Gapon convention involving anhydrous and hydrated reactions (Eq. 1 vs. Eq. 1, 3-4) overlapped for 0.01 mol L -1 concentrations (Figure 7), whereas a minor effect of salinity on the apparent selectivity coefficient was observed for 1 mol L -1 concentrations, which is the validity limit of the B-dot activity model [START_REF] Trémosa | Longterm assessment of geochemical reactivity of CO2 storage in highly saline aquifers: Application to Ketzin, In Salah and Snøhvit storage sites[END_REF]. Such discrepancy was due to the non-parallel evolution in water layer contributions from Na and Ca end-members with increasing ionic strength; as cation exchange and hydration reactions are coupled, clay hydration state affects in turn the exchange reaction through a feedback effect. This is supported by several works (e.g. [START_REF] Laird | Relationship Between Cation Exchange Selectivity and Crystalline Swelling in Expanding 2:1 Phyllosilicates[END_REF][START_REF] Van Loon | Mechanical Compaction of Smectite Clays Increases Ion Exchange Selectivity for Cesium[END_REF][START_REF] Whittaker | Ion exchange selectivity in clay is controlled by nanoscale chemical-mechanical coupling[END_REF]. Note that a concentration of 1 mol L -1 is equivalent to 95.6 and 94.2% of relative humidity in NaCl and CaCl 2 type solutions, respectively. Salinity range investigated thus corresponded to the domain where a high contribution of 3W was expected (Figure 4 and Figure 5). Despite 3W reactions being weakly constrained (see section 2.3), the exchange isotherms modelled at 0.01 and 1 mol L -1 reproduced the data, even for low salinity and high Na contents (Figure 7). Moreover, at high ionic strengths, formation of CaCl + ionic pairs and their incorporation in the interlayer space should ideally be taken into account (Ferrage et al., 2005b;[START_REF] Tertre | Influence of the ionic strength and solid/solution ratio on Ca(II)-for-Na+ exchange on montmorillonite. Part 1: Chemical measurements, thermodynamic modeling and potential implications for trace elements geochemistry[END_REF][START_REF] Tournassat | Influence of montmorillonite tactoid size on Na-Ca cation exchange reactions[END_REF]. As initial first approximation, the exchange of this complex was disregarded in our simulations. refer to the experimental data obtained by [START_REF] Tang | Cation-exchange kinetics on montmorillonite using pressurejump relaxation[END_REF] and [START_REF] Amrhein | Sodium-calcium exchange with anion exclusion and weathering corrections[END_REF] for the smectite SWy-1 at 0.01 and 1 mol L -1 , respectivelly.

Implications

The proposed approach does not require the thermodynamic properties of formation of the solids (i.e. clay layers) to be quantified, thus offering greater flexibility, and making it easier to include these models in complex, dynamic systems like soils and sediments. An issue hampering the inclusion of phyllosilicate hydration models in geochemical models resides in the fact that clays, and more especially smectite minerals, have exchangeable interlayer ions.

These exchange reactions influence in turn clay hydration behavior [START_REF] Bérend | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonites; 2, The Li + , Na + , K + , Rb + and Cs + -exchanged forms[END_REF][START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite; 3, The Mg 2+ , Ca 2+ , and Ba 3+ exchanged forms[END_REF]. Cation exchanges can be dealt with using non-ideal solid solutions including specific parameters such as Margules parameters [START_REF] Pabalan | Thermodynamics of ion exchange between clinoptilolite and aqueous solutions of na+k+ and na+ca2+[END_REF], but most studies dealing with cationic exchange use simpler [START_REF] Gaines | Adsorption studies on clay minerals. II. A formulation of the thermodynamics of exchange adsorption[END_REF], [START_REF] Gapon | On the theory of exchange adsorption in soils[END_REF][START_REF] Vanselow | Equilibria of the base-exchange reactions of bentonites, permutites, soil colloids, and zeolites[END_REF] formalisms, that involve fewer parameters. These models proved to be extremely robust and efficient for predicting cation migration in clay rocks [START_REF] Tournassat | Natural and engineered clay barriers[END_REF]. The developed approach to predict clay hydration is comparable to that established to quantify and predict cation exchanges in interlayer spaces and sorption on clay edges, which is known as 'surface complexation'. In addition, the developed model could be complemented with a surface complexation model to account for external water, based, for instance, on the work of [START_REF] Prost | State and location of water adsorbed on clay minerals; consequences of the hydration and swelling-shrinkage phenomena[END_REF] and/or [START_REF] Lindholm | Deconvolution of Smectite Hydration Isotherms[END_REF]. Such approach would allow refining water balance calculation in reactive transport modelling and coupling more accurately the feedback of chemistry on flow. The importance of such a coupling has been discussed by [START_REF] Seigneur | Recoupling flow and chemistry in variably saturated reactive transport modelling -An algorithm to accurately couple the feedback of chemistry on water consumption, variable porosity and flow[END_REF].

The coupling between hydration and exchange reactions could be also considered to study the evolution of apparent selectivity constants with the increase of the ionic strength (e.g. [START_REF] Liu | A cation exchange model to describe Cs+ sorption at high ionic strength in subsurface sediments at Hanford site, USA[END_REF]. Finally, the hysteresis of water sorption onto clays (e.g. [START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite. 1. The sodium-exchanged form[END_REF]) could be addressed using an energy barrier, the transition from a weakly hydrated species to a highly hydrated species (e.g. from 1W to 2W) then requiring more energy than the opposite reaction.

The consideration of an energy barrier could be similar to what is done for a mineral precipitation beyond an over saturation window. In water-saturated conditions and in presence of different cations (e.g. Na + and K + , the last one leading to weakly hydrated clay), this energy barrier could induce also an exchange hysteresis [START_REF] Laird | Relationship Between Cation Exchange Selectivity and Crystalline Swelling in Expanding 2:1 Phyllosilicates[END_REF], that remains to be thoroughly investigated.

Conclusion

The proposed modelling approach allows reproducing satisfactorily experimentally derived water vapor desorption isotherms as well as the different clay hydration states for contrasting charge deficits located in both tetrahedral and octahedral positions. Mass action equations derived from the Gapon convention are well-suited to describe clay hydration involving strong interactions between water molecules and the clay layer (i.e. tetrahedral charge). In contrast, the Rothmund-Kornfeld description may be used whatever the location of the layer charge deficit (tetrahedral or octahedral).

The proposed approach, based on an exchange model, is very easy to implement in geochemical codes and is satisfactory from a phenomenological point of view. Consistent with XRD characterizations of water vapor sorption isotherms, most of water molecules are found in smectite interlayers, whereas the relative proportions of the different smectite hydration states depend on both interlayer composition and relative humidity. The consideration of hydration reactions for the Wyoming bentonite does not indicate significant change on Na/Ca cation exchange within the concentration range studied (i.e. < 1 mol L -1 ) but further work is required to extend the model to other cations (K + , Mg 2+ , Sr 2+ etc.). In addition, this approach, by distinguishing explicitly the hydration/cation exchange process from the thermodynamic stability of the clay layer, is compatible with the use of reaction kinetics. In the long term, the method put forward here should allow linking chemistry to water content in soils, in clay rocks and mechanics that is swelling pressures.

Figure 1

 1 Figure 1. a) comparison of water contents derived from modelling of the XRD profile (red circles) with those determined from the water vapor desorption isotherms (black curve). Taken from Ferrage et al. (2010) for the saponite S-Na0.8. The green and blue curves indicate the quantity of water modelled for 1 and 2 layers (1W and 2W, repectively). The red curve represents the total quantity of water modelled (Total = 1W + 2W). b) evolution of the relative contribution of the different types of layers (0W, 1W and 2W) as a function of relative humidity for the saponite S-Na0.8. Symbols: experimental data from Ferrage et al. (2010). Curves: numerical results. Purple, green, and blue colors are respectively used for 0W, 1W, and 2W.

Figure 2

 2 Figure 2. a) comparison of water contents derived from modelling of the XRD profile (red circles) with those determined from the water vapor desorption isotherms (black curve). Taken from Ferrage et al. (2010) for the saponite S-Na1.4. The green and blue curves indicate the quantity of water modelled for 1 and 2 layers (1W and 2W, repectively). The red curve represents the total quantity of water modelled (Total = 1W + 2W) b) evolution of the relative contribution of the different types of layers (0W, 1W and 2W) as a function of relative humidity for the saponite S-Na1.4. Symbols: experimental data from Ferrage

Figure 6

 6 Figure 6. a) desorption isotherms obtained for the smectites Na-SWy-1, Ca-SWy-1 and Na/Ca-SWy-1. b) exchanger composition as a function of relative humidity for the smectite Na/Ca-SWy-1 (calculation performed without pore water, see text).

Figure 7 .

 7 Figure 7. Comparison of Na/Ca exchange isotherms obtained using anhydrous and hydrated reactions for total Na and Ca concentrations (i.e. [Na]+[Ca]) from 0.01 to 1 mol L -1 . The open circles and diamonds

Acknowledgments

The authors thank C. Tournassat for fruitful discussions, Carl I. Steefel for editorial handling and the anonymous reviewers for their constructive remarks. BRGM contributors acknowledge funding by internal research grants (HyDoLa and THERMODDEM projects). ISTerre is part of Labex OSUG@2020 (ANR10 LABX56). BL thanks the CNRS interdisciplinary research program Needs, through its "MiPor" project, for financial support.

Effect of smectite octahedral layer charge (hectorite)

In sodium hectorite, and in contrast to saponite, the relative contributions of 0W, 1W, 2W and 3W as a function of relative humidity are independent of amount of layer charge [START_REF] Vinci | Influence of layer charge on hydration properties of synthetic octahedrally-charged Nasaturated trioctahedral swelling phyllosilicates[END_REF]. As a consequence, hydration reactions must make use of equal exponents for water in the mass action equations independent of layer charge, and the Rothmund-Kornfeld description [START_REF] Bond | On the Rothmund-Kornfeld Description of Cation Exchange[END_REF] is probably the most straightforward for this purpose. The parameters were constant whatever the layer charge (Table 2) and allowed us to reproduce the experimental desorption isotherms and the contributions of 0W, 1W, 2W, and 3W for both H-Na 0.8 and H-Na 1.6 (Figure 3). An example of PHREEQC input file is given in Electronic Annex (see H-Na0.8.pqi). Note that the 3W contribution was significant only at high relative humidity, where the contribution of pore water was also strong. Fitted parameters for hydration reaction involving 3W layers are thus fraught with significant uncertainty.

Table 2. Reactions and selectivity constants used to simulate the dehydration of hectorites H-Na0.8 and H-Na1.6. For 1 g of H-Na0.8, a charge deficit of 0.8 eq mol -1 and a molar mass of 763 g mol -1 , the exchanger quantity is 10 -3 mol. For 1 g of H-Na1.6, a charge deficit of 1.6 eq mol -1 and a molar mass of 768 g mol -1 , the exchanger quantity is 2 10 -3 mol.

Clay

Effect of the type of interlayer cation (SWy-1 smectite)

Experimental desorption isotherms of the Na-exchanged SWy-1, hereafter referred to as Na-SWy-1 (Figure 4a - [START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite. 1. The sodium-exchanged form[END_REF], [START_REF] Bérend | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonites; 2, The Li + , Na + , K + , Rb + and Cs + -exchanged forms[END_REF]), were successfully modelled using the Rothmund-Kornfeld parameters listed in Table 3. These values allowed also reproducing the experimentally derived relative contributions of 0W, 1W, 2W, and 3W layers (Figure 4b). The water contents in 1W and 2W layers were estimated from the desorption isotherms. 3W layers were also considered in the modelling of the high relative humidity (Figure 4b), consistent with the reports of [START_REF] Holmboe | Porosity investigation of compacted bentonite using XRD profile modeling[END_REF] and [START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite. 1. The sodium-exchanged form[END_REF]. As for hectorite, parameters refined for exchange reactions involving 3W layers were poorly constrained owing to the overlap with the pore water contribution (Figure 4a). The same modelling procedure was applied to the experimental water desorption isotherm obtained for Ca-exchanged SWy-1 [START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite; 3, The Mg 2+ , Ca 2+ , and Ba 3+ exchanged forms[END_REF], hereafter referred to as Ca-SWy-1. As for Na-SWy-1, and although the plateaus in the desorption isotherms were less marked, a satisfying data modelling (with parameters listed in Table 3) could be obtained, both for the water desorption isotherm (Figure 5a) and for the evolution of the relative number of 0W, 1W, 2W, and 3W layers (Figure 5b). Overall, refined parameters (selectivity constants, Rothmund-Kornfeld coefficients, water contents for 1W, 2W, and 3W) differed significantly from that established for Na-SWy-1 (Table 3) due to the hydration behavior of Ca-SWy-1. In contrast to its Na-exchanged counterpart, 2W hydrates prevailed over a large range of relative humidity conditions in Ca-SWy-1. Table 3. Reactions and selectivity constants used to simulate the dehydration of smectites Na-SWy-1 and Ca-SWy-1. For 1 g of clay, a charge deficit of 0.74 eq mol -1 and a molar mass of 742 g mol -1 [START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite. 1. The sodium-exchanged form[END_REF], the exchanger quantity is 10 -3 mol.

Clay