Prof Wei

Elsheref Rodrigo Sigismund

Santos

Marie-Francoise Edwige

Zheng Zheng

Dawei Zhang

Tao Wang

Qingyi Gu

Ling Zhang

Junqing Gong

Xinnan Lu

Chen-Hui Chan

Xiaochen Lian

Cheng Yu

Jiawei Cai

Bingxu Zhang

Xiejun Chen

Xuezheng Zhao

Kaihang Shi

Zhengrui Li

Yuxiao Zhang

Hengbo Lv

Enyang Wang

Ph. D Committee

Prof Myroslav Holovko

Prof Chen Xiaosong

Prof

Prof Jing Jacques Mercadier

Prof. Yuanhui Xu

Prof Ji

Liang Qiao

Yan Wang

Si Wang

Limin Sun

Yinghua Wang

Huijun Zhou

Iii

Keywords: confined fluids, porous materials, statistical thermodynamics, adsorption, morphological thermodynamics V VIII

Porous materials are strategically important in chemical engineering, e.g., sustainable developments based on new clean energies, capturing Greenhouse gas, etc.

The development and application of porous materials involve studies of the thermodynamics and dynamics of fluids in porous materials. In last decades, plenty of experimental and theoretical studies have been reported. However, due to the variety of porous materials, and thermodynamic properties of confined fluid are affected by so many materials and fluid properties, e.g., pore size distribution, pore connectivity, etc., studies are still on a case-by-case way. The case-by-case study is hard to offer neither the control variables of confined fluid nor the common relation among the different confined fluids. The development of thermodynamic theories that can accurately describe the thermodynamic properties of confined fluids becomes more and more important. This thesis investigates the relation between confined fluid and the corresponding bulk fluid, interfacial properties of fluids at a curved surface, the general equation of state for confined fluids, and quench effect. This thesis is composed of following several parts: (1) With the help of scaled particle theory (SPT) and molecular simulation, we studied the thermodynamic properties of the confined fluid, e.g., pressure, chemical potential. A general scale relation has been found, which links the chemical potential, free energy density, and the pressure, from the confined fluids to that of the fluid in the homogeneous phase. This scale relation shows that the difference in thermodynamic properties between a confined fluid and a homogeneous fluid can be described solely by the porosity, the excess adsorption amount. The intrinsic relation between scaling relation and Gibbs adsorption theory is also revealed. This scaling relation provides a new method to measure the thermodynamic properties of confined fluid that are experimentally difficult to measure directly.

(2) By introducing a higher order curvature term into the SPT theory, we have developed a new, fully analytical approach called ASPT (augmented scaled particle theory). ASPT significantly improves the accuracy of SPT and gives excellent results for both the homogeneous phase and the surface tension of fluids at a spherical surface.

(3) Although morphological thermodynamics is supposed to be a general approach for studying confined fluids, it has never been tested in complicated systems, such as IV fluids in porous materials. We proposed an equation of state based on morphological thermodynamics for fluids confined in various situations. It turns out that this approach makes it possible to obtain excellent results for the thermodynamics of these confined fluids.

(4) The size of a medium is described by the chemical potential to insert a new particle. Using SPT theory and Monte-Carlo simulations, we have demonstrated that a quench-annealed system is more congested than a system without such disorders (e.g., binary mixture).

With the help of theoretical study and molecular simulation, this thesis studies the thermodynamic properties of confined fluids, clarifies the control variables of confined fluids, and discovers the common law of thermodynamics properties among different confined fluids. A scaling relation and two new equations of state were reported, they will deepen the understanding of confined fluid, and will advance the development of the thermodynamics for confined fluids.

Résumé

Les matériaux poreux ont une importance stratégique dans de nombreux domaines, par exemple, des développements durables basés sur de nouvelles énergies propres, des capteurs des gaz à effet de serre. Le développement et l'application de matériaux poreux impliquent des études sur la thermodynamique et la dynamique des fluides adsorbés dans des matériaux poreux. Au cours des dernières décennies, de nombreuses études expérimentales et théoriques ont été effectuées. Cependant, en raison de la diversité des matériaux poreux, les propriétés thermodynamiques des fluides confinés peuvent être affectées par de nombreux paramètres caractérisant les matériaux poreux, tels que la distribution de la taille des pores, la connectivité des pores, les interactions fluidematériau, etc. Jusqu'à présent, les études théoriques des fluides confinés ont été menées au cas par cas. Cette situation ne permet pas de voir clairement les caractéristiques communes des fluides confinés, encore moins les éventuels liens entre les fluides confinés et ceux en phase homogène. Lors de cette thèse, nous avons mené des études sur les quatre problèmes suivants : (2) En introduisant un terme de courbure d'ordre supérieur dans la théorie SPT, nous avons développé une nouvelle approche totalement analytique, baptisée ASPT (augmented scaled particle theory). ASPT améliore significativement la précision de la SPT et donne des excellents résultats à la fois pour la phase homogène et pour les la tension superficielle au voisinage d'une surface sphérique.

(3) Bien que la thermodynamique morphologique est sensée d'être une approche générale pour traiter des systèmes hétérogène dans lesquels une interface morphologiquement complique est présente, elle n'a jamais été testée dans des systèmes compliqués, comme par exemple des fluides confinés en milieux poreux. VI Nous avons mis au point une approche basée sur la thermodynamique morphologique pour des fluides confinés dans de diverses situations. Il s'avère que cette approche permet d'obtenir d'excellents résultats pour la thermodynamique de ces fluides confinés.

(4) L'encombrement d'un milieu est décrit par le potentiel chimique pour insérer une nouvelle particule. A l'aide de la théorie SPT et des simulations Monter-Carlo, nous avons démontré que un système avec des désordres figés (« quench-annealed » systèe) est plus encombré qu'un système sans tels désordres (par exemple, mélange binaire).

À travers les études décrites ci-dessus, nous avons réussi, à la fois, à développer certaines nouvelles méthodes théoriques pour des fluides confinés et à apporter des nouveaux éléments de réponse à quelques questions fondamentales qui étaient ouvertes auparavant.

Mots clés

Introduction 1.1 Porous material and its application

High-performance functionalized materials have strategic importance for many societal issues, e.g., sustainable developments based on new clean energies. Hydrogen is one of possible future clean energies. Scientific research has been actively undertaken to find a viable material for hydrogen storage [START_REF] Qiao | Connect the Thermodynamics of Bulk and Confined Fluids: Confinement-Adsorption Scaling[END_REF] . In the late 1990s, carbon nanotubes generated excitement for hydrogen storage while it turned out later that they do not attain the viable adsorption capacity, (more than 6% of the system's total weight).

Currently, the searching of such a storage system is directed toward new synthetic porous materials. [START_REF] Qiao | Fluids in porous media. IV. Quench effect on chemical potential[END_REF] Besides fluid storage, porous materials also have many other important applications. Various zeolites have been used as molecular sieves and catalysts. Due to the specific pore size and topology, high selectivity is imparted to the zeolite-based catalysts. Such selectivity is extensively exploited in various chemical processes, e.g., catalytic cracking in the oil industry. When chiral building blocks are used for elaborating porous materials, the selectivity can be pushed to a higher level and thus enantioselective porous materials have been synthesized [START_REF] Qiao | A Molecular Approach for Predicting Phase Diagrams of Ternary Aqueous Saline Solutions[END_REF] , which can have important applications in pharmaceutical industry. Now, material engineering allows for bestowing various interesting properties on porous materials. For example, magnetic porous materials have been synthesized [START_REF] 乔崇智 | 低浓度纳米颗粒在剪切 流中扩散、吸附的简单模型[END_REF] and some can have the transition temperature well above room temperature [START_REF] Qiao | Augmented Scaled Particle Theory[END_REF] . One attractive feature offered by the porous magnetic materials is their module ability. On one hand, the magnetic property can be modulated by choosing the appropriate building blocks for the synthesis. On the other hand, it can also be tuned, even after the materials being fabricated, by guest molecules adsorbed in the pores 6 . There is also considerable interest in the optical properties of some porous materials on account of their tunability and the possibility to incorporate a wide range of metal ions and organic ligand chromophores.

Such materials are of potential application as phosphors or fluorescent probes, particularly in chemical sensors [START_REF] Qiao | Virial Coefficient of a hard sphere fluid in random porous media[END_REF] . Another very interesting example is aerogel. Aerogels are synthesized by using a gas to replace the liquid in the pores of gel. Aerogels have many interesting properties, such as very large surface area, low density, large porosity, and very low thermal conductivity. Thus, the aerogel is an ideal thermal insulation material and has a potential application in aeronautics and astronautics. [START_REF] Yu | Enhancing Gas Solubility in Nanopores: A Combined Classical DFT and Machine Learning Study[END_REF][START_REF] Song | Interfacial Engineering of Thermoresponsive Microgel Capsules: Polymeric Wetting vs Colloidal Adhesion[END_REF][START_REF] Song | Membrane Wrapping Pathway of Injectable Hydrogels: From Vertical Capillary Adhesion to Lateral Compressed Wrapping[END_REF] Due to the large porosity and the very large surface area, aerogels are also wildly used in catalysis. [START_REF] Schneider | Aerogels in Catalysis[END_REF][START_REF] Quignard | Aerogel Materials from Marine Polysaccharides[END_REF][START_REF] Maldonado-Hódar | Catalytic Combustion of Toluene on Platinum-Containing Monolithic Carbon Aerogels[END_REF] The aerogel is a typically random porous material. With the development of synthesis technique, the ordered porous material such as Metal-organic frameworks (MOFs) and Zeolite imidazolate frameworks (ZIFs) have attracted more and more attention in both fundamental study and industry applications. For instance, MOFs is a coordination compound with a three-dimensional pore structure: organic struts link metal-containing clusters. MOFs were first synthesized in the late 1990s. Young 14 , Fujita [START_REF] Fujita | Preparation, Clathration Ability, and Catalysis of a Two-Dimensional Square Network Material Composed of Cadmium (II) and 4, 4'-Bipyridine[END_REF] , Venkataraman [START_REF] Venkataraman | Zeolite-like Behavior of a Coordination Network[END_REF] reported the applications of MOFs in catalysis of coordination polymers and gas adsorption. MOFs have many attractive properties i.e., low density, high solid-liquid interface area, and the controllable functionality. [START_REF] Schneemann | Flexible Metal-Organic Frameworks[END_REF][START_REF] Fortes | Negative Linear Compressibility and Massive Anisotropic Thermal Expansion in Methanol Monohydrate[END_REF][START_REF] Hodgson | Negative Area Compressibility in Silver(I) Tricyanomethanide[END_REF] The most interesting property of MOF is its flexibility. [START_REF] Bousquet | Adsorption Induced Transitions in Soft Porous Crystals: An Osmotic Potential Approach to Multistability and Intermediate Structures[END_REF][START_REF] Coudert | Adsorption Deformation and Structural Transitions in Metal-Organic Frameworks: From the Unit Cell to the Crystal[END_REF][START_REF] Krause | A Pressure-Amplifying Framework Material with Negative Gas Adsorption Transitions[END_REF] Normally, the supramolecular hostguest interaction leads to the flexibility of porous material. However, recent study found that the framework flexibility is also presented without the guest molecule and the phenomenon of adsorption or desorption. The framework flexibility of MOFs may be caused by some external stimulus in general (by guests and external force fields i.e., mechanical stress, photoresponse, thermoresponsivity, host-guest interaction, electrical and magnetic interaction). Due to the framework flexibility of MOFs, it has potential application in many industrial fields, such as gas separation, biomedical application, catalysis, and sensor. The MOFs sensor is based on the framework flexibility can cause a reversible color change. For instance, DUT-8(Ni) changes its color from yellow to green during the adsorption of CO2, n-butane, N2, etc. [START_REF] Yanai | Gas Detection by Structural Variations of Fluorescent Guest Molecules in a Flexible Porous Coordination Polymer[END_REF] Another example is based on the host-guest interactions, Yanai et al. were proposed that the composite material DSB@[Zn2(bdc)2dabco]n (DSB -distyrylbenzene) can be used to detect CO2 and C2H2. [START_REF] Klein | Monitoring Adsorption-Induced Switching by 129Xe NMR Spectroscopy in a New Metal-Organic Framework Ni2 (2, 6-Ndc) 2 (Dabco)[END_REF] Another very important application of porous materials is controlled releasing process of drugs. [START_REF] Koutsopoulos | Controlled Release of Functional Proteins through Designer Self-Assembling Peptide Nanofiber Hydrogel Scaffold[END_REF][START_REF] Horcajada | [END_REF] Considerable efforts devoted to developing methods for controlled drug-release to satisfy the ever-increasing demand for prolonged and better control of drug administration. The capability of continuous drug delivery over a specific period can assure an optimized therapeutic efficiency and a better patient's comfort for various chronic diseases, e.g., diabetes, cancers, AIDS, etc. A long release time requires a large capacity to store drug molecules. The controlled delivery necessitates some adaptability of the porous materials according to the drug content inside. New hybrid flexible porous materials opened some attractive perspectives for controlled drug delivery.

Above examples showed the strategic roles played by porous materials in a large variety of domains ranging from catalysis, separation, sensor technology to the pharmaceutical industry. It is estimated that porous solids represent more than 20% of the Gross Domestic Product of the industrial countries for the applications they imply directly or indirectly. [START_REF] Brinker | Porous Inorganic Materials[END_REF]

Experimental study on porous material

Since the porous materials have been used in many new areas, more precise control over the pore geometric properties is required. The templating-fabrication strategy is the most popular technique for control the pore size distribution and creates a variety of porous networks with a wide range of pore sizes from the micropore to macropore. 28- 33 In this method, Organic-bases molecules, polymer, and emulsion are normally considered as template species. The template species will be removed by heat treatment (soft template), acid or alkali solution (hard template) than the controllable pore scales can match the needs of different applications. Templating-fabrication strategy has widely used to synthesize ordered microporous material, such as MCM-48 and SBA-16, ordered mesoporous materials like MCM-41. Those materials are widely considered as catalysis, sensor, and capacitor. Another very interesting new synthesis technique is Pekala's method. In this method, a gas replaced the liquid in the pores of gel. Hence, aerogels have a network structure of interconnected nanosized primary particles and have a wide range pore size from micropores, which are related to the intra-particle structure, to macropores, which are produced by the inter-particle structure. Those two synthesis methods introduce how to control the pore structure, now, how to characterize the pore structure of random porous material will be introduced.

To characterize and reconstruct the morphologies of porous material, the statistical geometric properties obtained from two-dimensional images of microstructure, i.e. porosity, interfacial surface area, and the two-point correlation function are widely used. [START_REF] Adler | Real Porous Media: Local Geometry and Macroscopic Properties[END_REF] More recently the chord length distribution [START_REF] Shih | Image Processing and Mathematical Morphology: Fundamentals and Applications[END_REF] has been employed in the generation of 3D microstructural models. [START_REF] Roberts | Statistical Reconstruction of Three-Dimensional Porous Media from Two-Dimensional Images[END_REF][START_REF] Yeong | Reconstructing Random Media[END_REF] These methods have been very instructive in understanding the general properties of complex media, however, direct prediction of transport properties [START_REF] Adler | Flow in Simulated Porous Media[END_REF][START_REF] Adler | The Formation Factor of Reconstructed Porous Media[END_REF] from reconstructed samples have been in only fair agreement with experimental data. These standard methods base characterization and reconstruction primarily on geometrical information; there is no attempt to match the genesis of the material. Understanding how morphologies of porous material influence the properties of fluids in the porous material is important not only for a fundamental point of view but also for conceiving innovative and sustainable industrial processes.

Unfortunately, some thermodynamic properties of fluid in porous material still can not be obtained from the experiment, such as pressure and the free energy of the confined fluid. Hence, the theoretical study of the properties of fluids in porous material is still needed. In next section, some theoretical models will be introduced.

Theoretical Model

Although a large literature exists on the study of confined fluids by both experimental and theoretical methods. Today, we do not have yet a precise idea about the respective roles played by pore-connectivity, pore-size distribution, the morphologies of pore space, or quench disorder. The confined fluid particles feel not only the interaction with other fluid particles but also from solid ones. The interaction between fluid particles and porous materials can also make the confined fluid behave very differently from the bulk one. In this section, the models of pair potential and the models for porous material are introduced.

A. Models of the pair potential

Generally, the pair potential can be divided into two parts, the short-range repulsive part, which is derived from the overlap of outer electron shells, and the long-range attractive part, which comes from the spontaneous fluctuations of the electronic charge distribution. Due to the harsh repulsive part built the short-range order of fluid, which lead to the structure of fluid, the repulsive part is the most important property of the fluid. Hence, the first interaction model only considered the repulsion between fluid particles and the attraction is ignored, which is hard sphere potential. The pair potential, v(r), of hard sphere model is (1.1) where is the diameter of the fluid particle, and r is the distance between two fluid particles. Since only the repulsive part is considered in this model, which means the fluid only has one fluid phase, the vapor-liquid phase transition cannot be found in hard sphere fluid. And the freezing transition can be found in hard sphere fluid at high fluid density, . If the diameter, d, is equal to zero, the ideal gas model is obtained. In ideal gas model, the pair interaction is ignored, this model can be used to study the dilute system. Although this model is very simple, it still can offer many important features of confined fluid, as presented by Dong. 173 A simple pair potential model that can be used to describe the vapor-liquid phase transition can be proposed based on hard sphere model with an additional attraction, named square-well potential, ( )

,
where is the collective diameter, is the depth of the well. This model is widely used in molecular simulation. One should mention that this potential has a very long tail. To accelerate molecular simulation, a cut-off technique is widely used. To keep the continuity of the Lennard-Jones potential, it can be rewritten as . (1.4) And the Lennard-Jones potential always is divided by a reference hard sphere part and an attractive part, as Weeks-Chandler-Andersen theory 176 and Barker-Henderson theory 177 . From the above models of pair potential, one can found that the pair potential can always separate into two parts. This idea is from Van der Waals. In addition, perturbation theories, which is formed based on this idea, is a powerful tool for studying thermodynamic properties of fluids.

B. Models of porous material

Generally, the porous material can be divided into two parts, the ordered porous medium, and the disordered porous medium. The ordered porous medium like MOF or ZIF can be described by a repeatable unit cell. The disordered porous medium, random porous medium, are always built with the help of fixed particles. However, recent studies found that the structure of porous material might have changed, during the fluid adsorbed in, such as swelling, or contracting. [START_REF] Schneemann | Flexible Metal-Organic Frameworks[END_REF][START_REF] Fortes | Negative Linear Compressibility and Massive Anisotropic Thermal Expansion in Methanol Monohydrate[END_REF][START_REF] Hodgson | Negative Area Compressibility in Silver(I) Tricyanomethanide[END_REF][START_REF] Krause | A Pressure-Amplifying Framework Material with Negative Gas Adsorption Transitions[END_REF][178][179][180][181][182][183][184][185][186][187][188][189][190][191][192] In some cases, this behavior ( )

, , , 0 r v r r d r d s e s ¥ < ì ï = £ < í ï ³ î e d ( ) 12 6 4 v r r r s s e é ù ae ö ae ö = - ê ú ç ÷ ç ÷ è ø è ø ê ú ë û s e ( ) ( ) 12 6 4 c v r v r r r s s e é ù ae ö ae ö = - - ê ú ç ÷ ç ÷ è ø è ø ê ú ë û
cannot be ignored like in MOF or ZIF. [START_REF] Fortes | Negative Linear Compressibility and Massive Anisotropic Thermal Expansion in Methanol Monohydrate[END_REF]182,186 Most of models of disordered porous medium assumed that the structure of porous medium is a rigid one, which will not be modified by the behavior of fluid. This assumption makes the porous material can be considered as an external potential field, which is very convenient for theoretical approaches, such as, integral equations and density functional theory (DFT), and computer simulation. Although the structural change of porous medium is ignored, the description of the disorder of random porous material is still very complicated. The properties of fluid in random porous material must be taken two ensemble average. The first ensemble average is over all fluid configurations. And the second ensemble average is over all material configuration. Although the structure of random porous medium is a disordered and inhomogeneous one, after the ensemble average, it can be considered as an isotropic one. This concept is the fundamental basis of theoretical studies of Madden-Glandt model such as, Ornstein-Zernike equation 114,149,157,163 , Density Functional Theory [START_REF] Cheung | Quenched-Annealed Density Functional Theory for Interfacial Behavior of Hard Rods at a Hard Rod Matrix[END_REF]151,161,167,172 , and Scaled Particle Theory [START_REF] Chen | Comment on "A Highly Accurate and Analytic Equation of State for a Hard Sphere Fluid in Random Porous Media[END_REF][START_REF] Patsahan | Fluids in Porous Media. III. Scaled Particle Theory[END_REF][START_REF] Chen | Scaled Particle Theory for Multicomponent Hard Sphere Fluids Confined in Random Porous Media[END_REF][193][194][195][196][197][198][199][200][201][202] .

The first random porous medium model was proposed by W. G. Madden and E. D. Glandt in 1988. 40 In this model, the porous medium, also called matrix, is generated by fixing an equilibrium fluid, and the matrix particles are considered as an obstacle. The void among matrix particles is considered as pore. The interaction between matrix particles can be ideal gas potential or the hard sphere potential. And the porous medium is called an overlapping hard sphere (OHS) matrix and hard sphere (HS) matrix, respectively. And the interaction between fluid and matrix can be any pairwise additive potential. During the fluid adsorbed in porous medium, the fluid-matrix interaction will not modify the matrix structure. The matrix particle distribution is same as an equilibrium fluid distribution. With powerful theoretical tools, such as Ornstein-Zernike equation, density functional theory, numerous studies of confined fluid are carried out.

The behavior of fluids confined in random porous medium, such as, phase behavior [START_REF] Vink | Critical Behavior of Soft Matter Fluids in Bulk and in Random Porous Media: From Ising to Random-Field Ising Universality[END_REF][START_REF] Kalyuzhnyi | Phase Behavior and Percolation Properties of the Patchy Colloidal Fluids in the Random Porous Media[END_REF][145][146][147]150 , adsorption [START_REF] Links | Membrane Lateral Structure : The Influence of Immobilized Particles on Domain Size[END_REF][START_REF] Lomba | A Three Dimensional Integral Equation Approach for Fluids under Confinement: Argon in Zeolites[END_REF]104,105,110,124 , diffusivity [START_REF] Chávez-Rojo | Diffusion of Colloidal Fluids in Random Porous Media[END_REF]158 , were systematically studied.

However, many works of confined fluid are based on Madden-Glandt model, some fundamental questions are still unsolved. For instance, how to calculate the pressure of fluids in Madden-Glandt matrix is still questioned. Rosinberg et al. used a replica technique to calculate the pressure of fluid confined in Madden-Glandt matrix. [START_REF] Rosinberg | Thermodynamics of Fluids in Quenched Disordered Matrices[END_REF] And a complicated Virial expression for pressure was obtained. To obtain a simple expression of pressure in Madden-Glandt matrix, Dong also reported a simple expression of pressure which was obtained by considering the mechanic equilibrium. [START_REF] Dong | Mechanical Route to the Pressure of a Fluid Adsorbed in a Random Porous Medium[END_REF] In this same year, Kierlik et al. also proposed a so-called thermodynamics pressure, and found that this pressure and mechanical pressure are not identical. [START_REF] Kierlik | The Pressure of a Fluid Confined in a Disordered Porous Material[END_REF] the fluid pressure inside a random porous material. Also, the definition of the system volume may cause the difference among those pressure results. From an ideal gas system, one can find that if the system volume contains the volume occupied by matrix particle, the pressure equilibrium between the confined system and the corresponding bulk system will be broken. Hence, choosing the accessible volume as system volume seems more reasonable. However, the accessible volume is difficult to measure in experiment.

The morphology of some real porous materials like MCM-41 are sponge-like 203 , which is significantly different from the Madden-Glandt matrix. Hence, Zhao et al. 171 proposed a new matrix model, named hard sponge matrix model. In this model, the matrix particle is considered as the pore, which is not like the previous model where the matrix particle is considered as obstacle. This model can be considered as the opposite of overlapping hard sphere matrix. The Ornstein-Zernike equations of fluids confined in hard sponge model were proposed by Zhao et al. by using both diagram expansion and replica method. An analytical equation of state of fluid confined in hard sponge matrix was reported by Holovko et al. [START_REF] Holovko | Fluids in Random Porous Media: Scaled Particle Theory[END_REF] In this work, the strict derivation of this equation of state is leaked. Here, it should be noted that, in Madden-Glandt model, from the viewpoint of fluid, the interface between fluid and matrix is a concave one, in hard sponge model, the interface between them is a convex one. How this difference will change the thermodynamic properties of confined fluid will be discussed in Chapter 4.

Based on this model, Dong et al. [START_REF] Dong | Fluids Confined in Porous Media: A Soft-Sponge Model[END_REF] developed a new random porous model, called softsponge model, and the only difference from hard sponge model is when the fluid particle overlapped with material the interaction potential does not tend to be infinity, but finite positive value.

Above two porous medium models can describe some fundamental features of fluids in porous medium, such as disorder effect, the pore size distribution, and pore connectivity. In experiment, the templating-fabrication strategy skill is widely used in the synthesis of porous materials for controlling the pore size distribution.

Theoretical approach

Many interesting studies of the properties and phase behavior of confined fluids are based on some numerical methods, i.e., molecular simulation, density functional theory, and Ornstein-Zernike equations. In this section, two general methods which have potential to offer totally analytical results of fluids confined in different porous materials will be introduced in next two subsections. Some numerical methods for confined fluid will be also introduced in the last subsection.

A. SPT for confined fluid

Scaled particle theory (SPT) was first reported by Reiss et al. 204,205 in more than a half century ago. It provides a powerful tool to determine the equation of state of a bulk hard sphere (HS) fluid. SPT is not just limited to calculating the thermodynamical properties, i.e, pressure, chemical potential. It can also provide many surface and structural properties, i.e. surface tension, Tolman length, and radial distribution function.

Many extensions and applications have been reported, e.g. Dong and his coworkers [START_REF] Patsahan | Fluids in Porous Media. III. Scaled Particle Theory[END_REF]193,195 extend SPT to HS fluids confined in some random porous matrices. when the matrix density tends to zero and the matrix size tends to infinity, the expression of SPT1 cannot reduce to the original bulk SPT result. Later, with the help of a new formulation of SPT, this inconsistency was eliminated in SPT2. In this formulation, the morphology of porous materials is only presented by four parameters, packing fraction of matrix species, the ratio of the radio of fluid particle to matrix particle, and two porosities. The first porosity (geometrical porosity,

) is only about the geometry of porous materials which the probing particle is a point scaled particle.

The second porosity (probe particle porosity, ) measures the accessible volume of the center of fluid, and the probing particle is a fluid particle. The SPT2b shows a great agreement with the molecular simulation results. However, a divergence was found in SPT2b when the packing fraction of fluid is higher than the value of probe particle porosity. Based on the study of one-dimensional hard rod fluid in random porous materials [START_REF] Holovko | One-Dimensional Hard Rod Fluid in a Disordered Porous Medium: Scaled Particle Theory[END_REF] , an expansion of the term which contains probe particle porosity was used in SPT2b1 to eliminate this divergence. This expansion also improved the accuracy of the SPT prediction. However, it was found that when the fluid particle size is larger than matrix particle size, all versions of SPT significantly overestimated the thermodynamics properties of confined fluids even in fluid with a low density.

Comparing the expressions of SPT2, SPT2b, and SPT2b1, one can find that probe fluid particle porosity leads to this overestimation. The essence of those improvements of SPT is to reduce the influence of probe fluid particle porosity. Also, if we just use geometry porosity to replace probe particle porosity as SPT2a, the prediction of SPT will significantly underestimate the properties of confined fluid. Hence, find a more suitable porosity to describe the accessible volume of fluid is the key point to improve SPT. It should be mentioned that only four parameters are used for describing the porous material in SPT. We can assume that once those four parameters of porous material are obtained, the SPT can be used for describing the thermodynamic properties of fluid 0 f f confined in such porous material. Based on this, the SPT can be easily extended to other random porous materials, such as hard sponge matrices, and hard convex body matrices. 196,202 Later, the SPT was extended to study the thermodynamic properties and phase behavior of the hard-convex body fluid in random porous materials. 196,206 With the help of perturbation method, the phase behavior of network-forming fluid and ionic fluid. 194,201 Hence, the SPT can be considered as a general framework for fluids confined in different porous materials.

B. Morphological thermodynamics

Here, I introduce another general framework for confined fluid, morphological thermodynamics, which was proposed by Mecke and his co-works. [207][208][209][210] This method is from a beautiful mathematics theorem, Hadwiger's volume theorem. 211,212 It is not only a beautiful mathematics theorem but also offers a very powerful way to study confined fluid. Initially, Mecke and Wagner used this theorem to analyze the spatial patterns of galaxy distribution. 213 And then they found that this method can also be contributed to studies in microemulsions 214,215 , composite media 207,208 , complex molecule 216 . They named this method as morphological thermodynamics. In 2003, Bryk

217 derived an analytical equation to calculate the surface tension of a hard sphere fluid close to curved substrates by morphological thermodynamics. The result showed that the expression of surface tension of fluid close to arbitrary convex substrates does not include a logarithmic term of curvature. In 2004, König 209 derived a morphological thermodynamic form grand potential expression. This grand potential expression is used to calculate the solvation free energy of a cylindrical or a spherical hard sphere particle solute in hard sphere fluid.

( .5) where is the pressure and is the interfacial tension near a planar wall, and are the properties of the fluid. the volume, , the surface area, , the integrated mean curvature, , and the Euler characteristic, , are the geometric properties of

1 2 sol F pV A C X s k k = + + + p s 1 k 2 k V A C X solute.
The result obtained from the equation above has a good agreement with the result from Rosenfeld's fundamental measure theory. In 2009, Oettel 218 calculated the depletion potential between two hard sphere particles among the small hard sphere solvent using morphological thermodynamics and density functional theory (DFT). The result showed that morphological thermodynamics is invalidated when the distance between two hard sphere particles is greater than one solvent particle's diameter and smaller than a few of solvent particle's diameters, since the additivity restriction of Hadwiger's volume theorem is broken down in this situation. In 2012, Jin 219 studied the shape effect on solvation using morphological thermodynamics and DFT, the solvation free energy of solute with different shapes was calculated. Comparing with DFT result, it showed that when the solvent correlation length is smaller than the solute particle diameter, morphological thermodynamics is valid.

Recently, some studies [220][221][222][223][224] showed that the morphological thermodynamics is an approximate method but an exact one. Laird 223 used molecular dynamics simulation (MD) to calculate the interfacial tension, , between spherical or cylindrical particle and hard sphere fluid. They fitted the MD result to determine a polynomial of .

This result showed that the morphological thermodynamics is valid in low packing fraction ( ), while in a high packing fraction, the first non-Hadwiger coefficient showed a significant increase. Hansen-Goos 221 used Virial expansion to determine an exact surface tension expression, and this expression implied the existence of non-Hadwiger coefficient. To quantify the first non-Hadwiger coefficient, some simulations were carried out, the numerous results showed that the first non-Hadwiger coefficient is smaller than the smallest morphological thermodynamics coefficient in one order of magnitude. In those articles, they did not prove that the non-Hadwiger term coefficients with higher order are always in the same sign. In this case, the sum of non-Hadwiger terms may be equal to zero. In fact, their studies considered a fluid near spherical or cylindrical particle, it means that the interface between fluid and the obstacle particle is concave from fluid view, while the Hadwiger's volume theorem is only valid in the

g 1/ R 0.42 h < convex interface.
Although morphological is questioned recently, it is still considered as a potential tool for building a general framework for describing the fluid confined in different porous materials.

C. Other theoretical approaches

The Ornstein-Zernike equations and density functional theory are also the very popular theoretical methods for studying the thermodynamic properties of confined fluids. The two-body correlation functions are the basic quantities of Ornstein-Zernike equations. Since the replica Ornstein-Zernike equations of fluids in Madden-Gladnt random porous medium were reported by Given and Stell, [START_REF] Given | The Replica Ornstein-Zernike Equations and the Structure of Partly Quenched Media[END_REF] substantial effort has been focused on the study of the properties of a fluid confined in random porous materials using replica Ornstein-Zernike equations theory. For instance, Kierlik 106 used replica Ornstein-Zernike equations to investigate the phase diagrams of fluids in the random porous material, they found, in most cases, vapor-liquid coexistence curve of fluids in the random porous material is similar to the bulk ones, although it is narrowed. With the help of replica method, Zhao et. al. also proposed the Ornstein-Zernike equations for Hard-sponge and Templated random porous medium. However, the suitable closer for those Ornstein-Zernike equations is still not clear. Density functional theory is also a classical theoretical method for studying the inhomogeneous fluids, the porous material is generally considered as an external potential. For ordered porous material, density functional theory is widely used. For instance ,Liu et. al. 225 used density functional theory to predict the behavior of H2 adsorbed in different kinds of MOFs.

They found that the saturated pressure increases along with temperature in the low temperature region but a decrease in the high temperature region. For random porous material, if the material is still treated as an external potential, lots of configurations of material must be included in averaging procedure and the computation cost must be very expensive. Recently, Schmidt et al. 151,167 treated the random porous material on the level of one-body density distribution rather than as an external field and derived a so-called replica density functional theory.

Simulation method

Molecular simulation is a powerful and robust statistical mechanics method for confined fluid. [226][227][228] The basic idea of molecular simulation is to attain a set of configurations distributed according to some statistical distribution function, or statistical ensemble. There are two classical molecular simulation approaches for confined fluid: Monte Carlo method and molecular dynamics method. The molecular dynamics is based on the classical equations of motion or Langevin equation. Therefore, molecular dynamics can be used to study the time-dependent processes like diffusivity.

The Monte Carlo method is an essential stochastic method, where the configurations are generated by moving or inserting and removing particles. Therefore, to study the static properties of confined fluid, the Monte Carlo method will be more efficient. Many interesting studies of confined fluid are based on molecular simulation. For instance, the chemical reaction in porous medium, Turner [229][230][231][232] used Reaction ensemble Monte Carlo method to study reactions confined in porous material, they found the confinement effect is significantly improved the chemical reaction yield which almost cannot be observed in bulk system. Another interesting example is about phase transitions of fluids adsorbed in random porous material. Many experimental evidences support that the confinement effect can influence the liquid-vapor phase transitions. Such as comparing with the bulk vapor-liquid coexistence curve, the vapor-liquid coexistence curve of confined fluid is remarkably narrowed, and the critical temperature and density are lower. Since the properties of confined fluid are affected by many characteristics of porous material, such as porosity, pore size distribution, pore connectivity, how those characteristics of porous material will affect phase behavior had carried out a series of simulation studies. To accelerate the efficiency of the simulation of confined fluid, two powerful simulation techniques, the Gibbs-ensemble method, and the Gibbs-Duhem integration, were adapted by Brennan and Dong 153 .

In molecular simulation, the ordered porous medium like MOF and ZIF can be described by a repeatable unit cell. Since the disordered porous medium is always built with the help of fixed particles, for describing the structure of disordered porous material, lots of configurations (usually, about 5-20) of disordered porous medium are generated from the canonical ensemble Monte Carlo simulation, and the average over the material configurations was taken. Hence, the computation cost of computer simulation is expensive. In porous material, the pores can be divided into two parts, the open pores, and the closed (dead) pores. The open pores mean the fluid particle can access via diffusion process, while the dead pores are isolated from the outside fluid. In Monte Carlo simulation, the fluid particle can appear in closed pores via the random inserting operation, or via a large displacement operation.

Current problem and the objective of the present work

Although a large number of experimental and theoretical investigations have been made during the last decades, our understanding of confined fluids is still incomplete. Currently, we do not really know which extend thermodynamics can be applied at the scale of nanopores. Different aspects of confined fluids are being studied in a case-bycase way. In Chapter 2, we first time presented a general scaling relation between the confined fluid and bulk one-component fluid, which allows for connecting some thermodynamic properties of a confined fluid to bulk ones. Upon rescaling adsorbed fluid density, the adsorption-isotherms for many different confining environments collapse to the corresponding bulk curve. We also revealed the intimate connection of the reported scaling relation to Gibbs theory of inhomogeneous fluids.

As we mentioned before, morphological thermodynamics are questioned recently.

The existence of non-Hadwiger term has been confirmed by both theoretical study and molecular simulation. In the scaled particle theory, the formulation of the work for creating a spherical cavity in a fluid is identical to the morphological thermodynamics, and the scaled particle theory for the bulk system is not as accurate as other equation of state such as Carnahan-Staring equation of state. This indicated that the scaled particle theory can be improved by adding a non-Hadwiger term. In Chapter 3, we set up two new versions of SPT. The first one contains an adjustable parameter. The second one uses two Laurent series to describe the chemical potential for inserting a scaled particle.

Both new versions of SPT significantly improved the accuracy of not only thermodynamic properties, i.e. pressure and chemical potential, but also surface properties, i.e. surface tension and Tolman length. We also first time obtained a selfconsistency expression of non-Hadwiger term. Moreover, the same idea can be used to treat the multi-components system, and the improvement is significant.

Morphological thermodynamics offered a tool for building a general framework to describe the thermodynamic properties of fluids confined in different porous materials.

However, how to use this tool is still unclear. In Chapter 4, by considering the chemical potential as the independent variable, we derived, for first time, a new general equation of state by using morphological thermodynamic. In this equation of state, the porous material is described by only four geometric properties, i.e., the geometry porosity, fluid-solid interface area, integrated Gaussian curvature, and integrated mean curvature.

To our best knowledge, this is the first equation of state for confined fluid which is irrelevant to the model of porous medium. Our new equation of state has a great agreement with molecular simulation results in a large range.

Madden-Glandt model is the first model of random porous material. In this model, the disordered porous material is mimicked by random matrix configuration that can be generated by fixing a relevant fluid system at its equilibrium state. Thus, the quenched "fluid" particles constitute the matrix (thereafter called matrix particles), and the voids among them are pores. In Chapter 5, we proposed a quantitative measure of quench effect for the adsorption of fluids in random porous materials. With the help of scaled particle theory and molecular simulation, we find when matrix particles are larger than fluid particles, the confined fluid shows the same behavior, i.e., chemical potential, as the binary mixture bulk system.

A confinement-adsorption scaling relation 2.1 Introduction

Accompanying the elaboration of high-performance functionalized nanoporous materials, a large number of experimental and theoretical investigations have been made during the last decades. Nevertheless, our understanding of confined fluids is still incomplete. Currently, we do not really know to which extent thermodynamics can be still applied at the scale of nanopores. Different aspects of confined fluids are being studied in a case-by-case way. It can be readily admitted that the fluid-solid interface and the fluid inhomogeneity near it have to be taken into account. For fluid adsorption in the real porous material, the fluid-solid interfaces are generally curved ones. It might appear surprising that the thermodynamics for dealing with curved interfaces is not so well established although early investigations go back to Tolman. 233 Mecke and coworkers have made efforts to develop a general framework, named as morphological thermodynamics, to account for more complex surface morphology. 209,210,217,220,234 The foundation of morphological thermodynamics has been questioned recently. [221][222][223][224] To our best knowledge, no experimental measurement has ever been made to determine the bending rigidity coefficients needed in morphological thermodynamics method for any fluid-solid interface.

A large literature exists on the study of confined fluids by theoretical and simulation methods. Models with simple pore geometry (e.g., slit or cylinder) are widely studied.

In such models, the pore-size distribution and connectivity among pores are neglected.

Fluid adsorption and diffusion in ordered porous material, e.g. zeolites, have been studied by simulations. To account for the quenched disorder, models for random porous media have been proposed also, e.g., Madden-Glandt model and various variants. [START_REF] Madden | Distribution Functions for Fluids in Random Media[END_REF][START_REF] Dong | Integral Equations for a Fluid near a Random Substrate[END_REF]108,136 Despite these considerable efforts, it is unfortunate to note that no unifying picture of various confined fluids has emerged. Today, we do not have yet a precise idea about the respective roles played by pore-connectivity, pore-size distribution, pore morphology, or quenched disorder. In a bulk fluid, a molecule is surrounded by other fluid molecules while in a fluid adsorbed in a porous solid, a large number of fluid molecules are located near a fluid-solid interface. These molecules feel the interaction with both fluid and solid molecules. The nature of fluid-solid interaction can vary significantly, from repulsive to attractive ones. This additional interaction can make the confined fluid behave very differently from the bulk one. Although a confined fluid appears complicated due to the complex confining environment of adsorbent, one can wonder if there is any connection between confined and bulk fluids. Currently, we know quite a few about this. Acquiring such knowledge does not only advance our understanding of these complex systems but also can have important applications. In this chapter, we report several relations which allow for connecting some properties of a confined fluid to those of a bulk one. By rescaling the density of a confined fluid, the adsorption isotherms (also free energy or grand potential per particle) of fluids in a large variety of confining environments can collapse to the corresponding bulk ones.

Model and method

We investigated the thermodynamic properties (chemical potential, Helmholtz free energy, and grand potential) of confined fluids by considering a large variety of models (hard-sphere or Lenard-Jones fluids in slit-pores, ordered or disordered porous matrices). Simulations were carried out with the help of Monte-Carlo methods (in grand-canonical or canonical ensembles).

A. Model

A-1. HS fluid confined in various porous environments

In the present work, we consider only one-component fluid (denoted as species 1).

The fluid-fluid interaction between hard spheres of radius, , is given by

1 R . (2.1)
where and are the position vectors of the ith and jth fluid particle respectively.

Various confining environments are considered. For Madden-Glandt model of random porous matrices (denoted as species 0), the following fluid-matrix interaction is given by (2.2)

where is matrix particle radius, the position vector of the jth matrix particle, and are respectively the potential-well depth and width. In the case of and , we have a hard sphere (HS) matrix. The configurations of an HS matrix are generated from an equilibrium system with the following interaction, .

For an overlapping hard sphere (OHS) matrix, matrix particles are placed totally randomly, i.e., . We considered also a slit pore with the width of and the interaction between fluid and the pore wall is given by , (2.4) where and is the coordinate along the coordinate axis perpendicular to the slit walls (note that the origin of the coordinate system is placed at the middle of the slit pore). In the case of and , we have the simple case of a slit pore with two hard walls. For slit pores, we calculate the fluid density by using the physical volume, i.e., ( : surface area) but not the volume accessible to the centers ( )

1 11 1 2 0 2 i j i j i j R u R ì ¥ -< ï - = í -³ ï î r r r r r r i r j r ( ) 0 1 10 0 0 1 0 1 0 1 , 0 j j j j R R u R R R R d R R d e ì ¥ -< + ï ï - = + £ -< + + í ï + + £ - ï î r q r q r q r q 0 R j q 0 e d 0 0 e = 0 d = ( ) 0 00 0 2 0 2 i j i j i j R u R ì ¥ -< ï - = í -³ ï î q q q q q q ( ) 00 0 i j u - = q q L ( ) ( ) ( ) 1 0 1 2 2 2 0 2 i i i i z L w z L d z L z L d s e s ì¥ ³ - ï = -£ < - í ï < - î 1 1 2R s = i z 0 0 e = 0 d = V AL = A of fluid particles.

A-2. LJ fluid confined in various porous environments

To demonstrate the validity of the scaling relation when an attractive fluid-fluid interaction is also present, a Lenard-Jones fluid with the following interaction is considered as well, ( , 0 , is calculated for different density, ( , : the number of fluid particles), by using Widom's test particle method. 226 Then, Helmholtz free energy is determined by a thermodynamic integration, i.e., (2.11) where is the thermal wavelength of fluid particles and in this work, we set .

c c i j i j i j c i j u u r r u r ì - - -£ ï - = í -> ï î r r r r r r r r ( ) ( ) ( ) 12 6 LJ 1 1 1 =4 , u r r r e s s é ù - ë û 11 1 10 c r R = 1 1 2R s = 1 e ( ) ( ) ( ) LJ LJ 10 10 10 10 , 0 c c i j i j i j c i j u u r r u r ì - - - £ ï - = í - > ï î r q r q r q r q ( ) ( ) ( ) 12
Finally, we obtain readily compressibility factor from the following thermodynamic relation,

(2.12)

C. Conditions of considered systems

The computational conditions of all the considered systems are summarized in Table 2.1. The confining environments considered in this work can be classified into two big categories: i) porous matrices and ii) slit pores. According to their different morphologies of pore space, we can divide porous matrices into four different types.

Disordered porous matrices are generated by quenching an equilibrium system according to the procedure proposed by Madden and Glandt [START_REF] Madden | Distribution Functions for Fluids in Random Media[END_REF] and denoted, in Table 2.1, as HSM_d if matrix-matrix interaction is HS one or LJM_d when the matrix-matrix interaction is LJ one. In contrast, ordered porous matrices can be generated by arranging matrix particle into an ordered structure. In this work, we studied only the case that

L 1 ex µ 1 r 1 1 / N V r = 1 N ( ) ( ) 1 3 1 1 0 1 1 1 ln -1 d , ex F N r b r b µ r r r = L + ò 1 L 1 1 s L = 1 1 . F N N b b bµ W = -+
matrix particles are places on a simple cubic lattice and the acronyms used for such matrices are HSM_o (HS for fluid-matrix interaction) and LJM_o (LJ for fluid-matrix interaction. The third type of matrices we considered, is templated matrix. Following the procedure proposed by Van Tassel et al, 108,136 a templated quenching an equilibrium binary system and removing its one component after quenching. We considered only templated HS matrix, denoted by THSM in Table 2.1, with the same number of template and matrix particles and moreover they have the same size. The fourth type of matrices is the hard sponge one, [START_REF] Holovko | Fluids in Random Porous Media: Scaled Particle Theory[END_REF]171 denoted as HSG in Table 2.1.

The meaning of the title labels in Table 2.1 is the following, f-s: fluid-solid interaction (here the word "solid" is used to denote either hard wall or matrix particles); f-f: fluid-fluid interaction; τ: fluid matrix particle size ratio, ;

: temperature, ( : reference energy unit, all the well-depth parameters for square-well or LJ potentials, e.g., , , are defined with respect to this reference unit);

Symbol: symbols used for curves plotted in different figures of the paper.

The definitions of all the other reduced parameters given in Table 2.1 are given below as well.

Slit width: ; SW( , ): square-well potential with width, , and depth, ;

LJ( ): LJ potential with depth, ;

: matrix density, ( , : number of matrix particles).

Results and discussion

Inspired by the exact and analytical results for some models (ideal gas in a variety of confining environments) and scrutinizing our simulation results for many more complex confined fluids with interactions, we found the following scaling relation:

, (2.13) , (2.14)
, (2.15) where ( : Boltzmann constant, T: temperature), and are respectively chemical potential of confined and bulk fluid, the number density of the bulk fluid in equilibrium with the confined one ( , : volume), and are respectively free-energy per particle of confined and bulk fluid, i.e., , ( : particle number of confined fluid), grand-potential per particle of a confined fluid, is the compressibility factor in the bulk, i.e., , ( : bulk fluid pressure, note that the compressibility factor is simply the negative of grand-potential per particle). The scaling factor is given by: ,

where is geometric porosity of the porous adsorbent under consideration (see [START_REF] Holovko | Fluids in Random Porous Media: Scaled Particle Theory[END_REF]

( ) ( ) , , bulk T T bµ cr bµ r = ( ) ( ) , , bulk f T f T b cr b r = ( ) ( ) , , bulk T Z T bw cr r - = 1/ ( ) B k T b = B k µ bulk µ r / N V r = V f bulk f / conf f F N = / bulk bulk f F N = conf N w bulk Z / bulk bulk Z P b r = bulk P 0 / ( ) bulk A P V c f b = +G 0 f G A A G 0 ( ) conf A V r f r G = - / conf conf N V r = A G G A A G
hand side of eq. ( 2.16) is dropped out, we have immediately: ;

(2.18)

; (2.19) . (2.20) We name this as pure-confinement scaling which holds rigorously for an ideal gas in hard matrices, e.g., a hard-sphere (HS), an overlapping HS (OHS) or a hard-sponge matrix. From scaled particle theory, [START_REF] Chen | Comment on "A Highly Accurate and Analytic Equation of State for a Hard Sphere Fluid in Random Porous Media[END_REF][START_REF] Patsahan | Fluids in Porous Media. III. Scaled Particle Theory[END_REF][START_REF] Holovko | Fluids in Random Porous Media: Scaled Particle Theory[END_REF][START_REF] Chen | Scaled Particle Theory for Multicomponent Hard Sphere Fluids Confined in Random Porous Media[END_REF]193,195 we can show that the pureconfinement scaling holds also for HS fluids confined in HS and OHS matrices when the size ratio of matrix to fluid particle is large. 193 The simulation results presented below demonstrate the validity of the scaling relation given in eq.( 2.13)-( 2 ( ) ( ) a derivation of these relations from first-principles is currently lacking. Nevertheless, the confinement-adsorption scaling has some intimate connection with Gibbs theory for interfacial systems 236,237 and the morphological thermodynamics advocated by Mecke et al. 209,210,217,220,238 In fact, Gibbs theory can be derived from the scaling relation given in eq. (2.13). We start from the following equivalent form of eq. ( 2.13), .

0 , , bulk T T bµ f r bµ r = ( ) ( ) 0 , , bulk f T f T b f r b r = ( ) ( ) 0 , , bulk T Z T bw f r r - = μ r
b bw - f b bw - ( ) ( ) , , bulk f T f T b µ b µ = ( ) ( ) , , bulk T Z T bw µ µ - =
(2.23)

Expanding the RHS of this equation to the first order around , we obtain, (

Eq. ( 2.17) was used when going to the last equality of eq. (2.24). Integrating both sides of the above equation with respect to leads to, (2.25) where is the surface tension at fluid-solid interface and the surface area of the pore space boundary used for calculation the porosity, . Gibbs adsorption equation was used when going to the last equality in the above equation. Eq. (2.25) is nothing else but the free energy of the inhomogeneous system expressed as the bulk contribution plus the surface term following Gibbs theory, i.e., (2.26) ( )

, , conf conf bulk T T r bµ r bµ c ae ö = ç ÷ è ø 0 / r f ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 0 0 , , , , , , , bulk conf conf conf conf conf bulk conf bulk conf conf conf bulk conf bulk conf conf bulk conf T T T T T T A T V bµ r f r r r bµ r bµ f c f r f bµ r f r r bµ r f f r f bµ r f r bµ f f r é ù ¶ ae ö ae ö ë û = + - ç ÷ ç ÷ ¶ è ø è ø é ù ¶ ae ö ae ö ë û = + - ç ÷ ç ÷ ¶ è ø è ø é ù ¶ ae ö G ë û = - ç ÷ ¶ è ø ( ) 0 f conf r ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0 0 0 0 , , , , , , , ,
conf conf conf bulk conf bulk conf conf conf bulk conf F N V T T d V T F N V T A d V V F N V T A V V b bµ r r bµ r f b f f rf f r f b f bg f f = é ù ¶ G ë û = - ¶ = + ò ò g A 0 f ( ) ( ) 0 , , , , conf bulk conf F N V T F N V T A f g = +
Subtracting from both sides of the above equation, we obtain also:

( 2.27) where , denote respectively the grand potential of confined or bulk fluid.

According to morphological thermodynamics, the grand potential of an inhomogeneous fluid is written as the sum of a bulk term and a surface term with the surface tension including three contributions (flat surface term plus two curvature terms). [START_REF] Zhao | Templating Methods for Preparation of Porous Structures[END_REF][START_REF] Raman | Template-Based Approaches to the Preparation of Amorphous, Nanoporous Silicas[END_REF][START_REF] Yuan | Insights into Hierarchically Meso-Macroporous Structured Materials[END_REF][START_REF] Ma | A Review of Zeolite-like Porous Materials[END_REF][START_REF] Adebajo | Porous Materials for Oil Spill Cleanup: A Review of Synthesis and Absorbing Properties[END_REF] Although not appearing explicitly in our scaling relation, the surface tension is embodied in it.

The above demonstration reveals this unambiguously and thus evidences an intimate connection between our scaling relation and the general theoretical frameworks like Gibbs theory and morphological thermodynamics. At first sight, it may appear surprising that rescaling alone fluid density can account for various fluid-solid interaction. In fact, this interaction is taken into account through the adsorption term, i.e., [see eq.( 2. 16)]. The discussion just given above shows further and explicitly that the interface contribution to the free energy is indeed included in our scaling relation.

Scaling relations have been found previously for some dynamic properties, e.g., entropy scaling for the diffusion coefficient of an HS fluid confined in slit pores 168,239,240 or an LJ fluid in a zeolite, 241 or for the relaxation time of a glass-making liquid in slit pores. 242 Mittal, Errington, and Truskett found that the ( : diffusion coefficient; : excess entropy per particle) curves of an HS fluid confined in various slit pores collapse to the bulk curve when the fluid density is calculated with the total volume instead of that accessible to the particle centers. 168 This way to obtain curvecollapse is, in fact, a particular case of the general scaling reported in the present work.

First, the situation considered by these authors corresponds to what we called pureconfinement regime, i.e., without the second term of the scaling factor given in eq.

(2.16). Our investigations show that the general scaling relation holds under wider conditions. Second, applying eq.( 2.16) to the particular case of slit pores under the pure-

conf N µ ( ) ( ) 0 , , , , bulk V T V T A µ µ f g W =W + W conf W A G ex D s ! D ex s
confinement regime consists simply in calculating the fluid density by using the total volume as did Mittal,Errington,and Truskett. 168 In a later study, Mittal showed that data collapse to the bulk curve can be obtained also if the diffusion coefficient is plotted as a function of compressibility factor. 242 Our finding reported here provides the thermodynamics foundation for this. It is to note that in the case of an HS fluid, the reduced free energy per particle, , is equal to the reduced entropy per particle, . So, our results are perfectly consistent with those of Mittal et al. 168,239,240 Moreover, in light of our finding, we can make an immediate prediction that curves for confined fluids collapse also to the corresponding bulk one.

Conclusion

Our simulation results establish the validity of a scaling relation for several thermodynamic functions which connects confined and bulk fluids. The invariance with respect to confining environments is discovered for Helmholtz free energy and grandpotential per particle if they are expressed as a function of chemical potential. This confers a particular significance to the use of chemical potential as an independent variable in the study of confined fluids or inhomogeneous fluids. The invariance described by eqs. (2.21) and (2.22) holds rigorously for an ideal gas confined in various pores under all the allowed thermodynamics conditions. It is really surprising that such invariance holds also when fluid-fluid interaction is present. Fig. 2.4b shows pretty good data-collapse in the density region where the compressibility factor deviates largely from its value for an ideal gas. It is very intriguing that confined fluids can bear perfectly a hallmark of an ideal gas far beyond the low-density region. Although a derivation of the scaling relation from first-principles is currently unavailable, we have revealed its intimate connection with general theoretical frameworks like Gibbs theory or morphological thermodynamics for inhomogeneous fluids. We believe this is why the scaling relation works so well under wide conditions and for a large variety of confining environments. The most significant message conveyed by our results is that

f b / B s k D µ !
the apparently disconnected behaviors of confined fluids are not so disparate but can be nicely organized via scaling. The scaling relation shows clearly that the porosity (space accessibility) and fluid-solid interaction (through the adsorption term) is of primary importance for determining the thermodynamics of confined fluids. The other characteristics, like pore-connectivity, pore-shape, pore-size distribution, etc., play a less significant role. As an immediate and interesting application, our finding allows for circumventing some experimental difficulty for direct determination of some thermodynamic properties of confined fluids. A challenge in perspective is to see if the scaling relation holds also for the fluid adsorption in flexible porous materials, e.g., metal-organic frameworks (MOFs) or to find the modifications needed if necessary.

Development of augmented scaled particle theory

Introduction

Worked out by H. Reiss, H. L. Frisch and J. L. Lebowitz in 1959, 204,243 scaled particle theory (SPT) has become one of the most successful theories in liquid physics with widespread applications. Since then, there have been continuous efforts to extend and improve it (see, e.g. [START_REF] Patsahan | Fluids in Porous Media. III. Scaled Particle Theory[END_REF][START_REF] Holovko | One-Dimensional Hard Rod Fluid in a Disordered Porous Medium: Scaled Particle Theory[END_REF][START_REF] Holovko | Fluids in Random Porous Media: Scaled Particle Theory[END_REF][START_REF] Chen | Scaled Particle Theory for Multicomponent Hard Sphere Fluids Confined in Random Porous Media[END_REF]195,[243][244][245][246][247][248][249][250][251][252][253][254] , which is by no means an exhaustive list of the very large number of papers on SPT in the literature). The original motivation for SPT is to derive a simple equation of state (EOS) for a bulk hard sphere (HS) fluid.

Then, it was recognized that SPT provides not only the thermodynamic properties of an HS fluid in bulk but also the surface tension of an HS fluid near a spherical hard wall.

Within the framework of SPT, this surface tension contains three terms, i.e., the surface tension at a flat hard wall and two contributions due to surface curvatures (mean and Gaussian curvatures). Investigations of more and more complex inhomogeneous systems, e.g., fluids adsorbed in various porous materials, arouse increasing interests in interfacial properties. Although early investigations go back to Tolman, 233 it might appear surprising that our knowledge about curved interfaces is still quite limited.

However, the accuracy of its predictions is not as high as other recent equation of state like Carnahan-Starling (CS). 255 To obtain an equation of state from SPT, a form of the reversible work of inserting a large hard sphere potential or a form of the central function G must be presumed. Normally, the reversible work is presented by a Laurent series, ,

where is the pressure of the hard sphere fluid, is the coefficient and is the radius of the inserted particle. In original work, N = 2 in Laurent series and three exact conditions of the reversible work was used to solve those three coefficients, the pressure ( ) Mecke and co-workers have made efforts to develop a general framework, named as morphological thermodynamics, to account for more complex surface morphology. [208][209][210]224,257 According to Hadwiger theorem 211,212 in integral geometry, the morphological thermodynamics postulates that the thermodynamic potential of an inhomogeneous system is determined only by four terms, i.e., one bulk contribution proportional to system's volume and three surface contributions proportional respectively to surface area, mean and Gaussian curvatures of the interface. It is to be pointed out that in SPT, the chemical potential for creating a spherical cavity inside a fluid is assumed to have this form. The foundation of morphological thermodynamics has been questioned recently. [220][221][222][223][224]

3 2 0 4 3 N i s s is i W R P R
1 (2 / 3 1) s R R > - 1 R s R
improved the predictive accuracy of both thermodynamical properties and surface properties, however, this improved SPT cannot offer an analytical equation of state, since an integral equation is contained.

From these recent advances, it is clear now that the thermodynamic expression for chemical potential in the original SPT does not account for the contribution from non Hadwiger terms. One can raise naturally the following questions. Is it possible to include some non Hadwiger terms in SPT? How this can be carried out in a simple way, e.g., obtaining still analytic and improved results? We address these issues in the present work.

Original SPT

A brief recall of SPT is given first to introduce some notations and formulas needed in the following section. For the chemical potential to insert a hard scaled particle with a radius, (a variable one), into a fluid of hard spheres of radius, , the following exact but formal result is well known, 204

(3.2)
where is the thermal wavelength, ( : Boltzmann constant, : temperature), is the system volume, and is the number of hard spheres. We consider here the case that the scaled particle has the same mass as the HS particles already in the system and so they have the same thermal wavelength. The scaled particle becomes indistinguishable from the other fluid particles only when . In eq. ( , ) 1 ( ) ln ln ( ) 1 ( )

s s s s N R R R pR V bµ D + ae ö = L - ç ÷ è ø L kT / 1 = b k T V N s R R = 1 ( , ) 0 s s s R R R R R R = ì D = í ¹ î ) ( 0 s R p
s n s n p R F R ¥ = = + å ( ) ( ) 1 2 1 2 1 1 ( ) ... ( , ,..., ) ! n n n s s n n s i i F R d d d d f n V r = = - Õ ò ò ò ò r r r r r r r r r ( ) 1 2
( , ,..., )

n n r r r r n 1 2
, ,..., n r r r

( ) ( ) 1
s s i u s i f e b - - -= - r r r r ( ) 0 s i s s s i s i s R R u R R ì¥ -< + ï - = í -³ + ï î r r r r r r 3 1 ( ) 1 s s s R F R R R R h ae ö = - + > - ç ÷ è ø ( ) 2 2 2 2 (2) 2 2 0 0 ( ) 2 () , 0 s s R R s s s R R R F R drr r r R R p r + -< < ì ï = í W ³ ï î ò ( ) ( ) 0 3 2 3 3 n s s F R n R R R = ³ -< £ - 3 4 /3 R h pr = / N V r = ( ) 1 2 W - r r s R R + 1 2 - r r ( ) ( ) 2 1 2 1 2 2 1 2 1 2 2 ( ) 2 , 0 2 3 2 2 s s s R R R R R R p ae -ö é -ù W - = + - + + < -£ + ç ÷ê ú è øë û r r r r r r r r
infinite series and we do not know how it converges for an arbitrary value of .

Moreover, only has an analytical expression and it is more and more difficult to determine the other higher order terms. For , we only know its first three derivatives are continuous at .

The strategy used for deriving the original SPT is to use the above exact result in a limited range of , i.e., . (3.12) For larger values of , the following thermodynamic expression is proposed, ,

Where P is the pressure of the HS fluid and the terms involving lower powers of account for the surface tension around the scaled particle with being related to the flat surface contribution and , to the respective contributions from the mean and Gaussian curvatures. In eq. (3.13)

, (3.15) . (3.16) The chemical potential, , and the pressure, , can be obtained from, , (3.17

) s R 1 ( ) s F R 2 ( ) s F R ( ) 2 3 3 s R R = - s R 3 3 ( ) ln ln 1 1 0 s s s s R R RR V R bµ h é ù ae ö L ae ö = - - + -< £ ê ú ç ÷ ç ÷ è ø ê ú è ø ë û s R 3 3 2 0 1 2 ( , ) 1 4 1 ( ) ln 0 2 3 s s s s s s N R R R P W R w w R w R R V p b b D + ae ö = L + + + + > ç ÷ è ø s R 2 w 1 w 0 w ( ) 1 n s R 1, 2,3,... n = 0 s R = ( ) 0 ln 1 w h = - - 1 3 1 1 w R h h = - h h h h ae ö = + ç ÷ - - è ø µ P ( ) s W R R bµ b = =
and Gibbs-Duhem equation, i.e., . (3.18) Finally, one obtains, ,

, (3.20) for the pressure and chemical potential of a bulk HS fluid. Moreover, one obtains also the following dimensionless result for the surface tension of an HS fluid on a spherical hard wall of radius, , ,

where is the SPT result for the surface tension of an HS fluid on a flat hard wall, ,

and , account for respectively the contributions to the surface tension from mean and Gaussian curvatures which have the following expressions within the framework of SPT, ,

.

(3.24)

Augmented SPT

A. Single-point matching formulation

A conceptually simple strategy for including the leading non Hadwiger term in an ( ) ( )

P b bµ r r r ¶ ¶ = ¶ ¶ ( ) 2 3 1 1 SPT P b h h r h + + = - ( ) ( ) 2 3 6 9 ln ln 1 1 2 1 SPT SPT P h h b bµ r h h h h r ae ö = L - -+ + + ç ÷ - - è ø s R ( ) ( ) 2 2 0 1 2 4 SPT SPT SPT SPT SPT s s s s R R R R R R R g p b g g g g - - ae ö = = + + ç ÷ è ø ! ! ! ! 0 SPT g ( ) 2 2 0 3 3 2 2 2 1 1 SPT w R h h g h h ae ö = = + ç ÷ - - è ø ! 1 g - 2 g - 1 1 3 1 SPT w R h g h - = = - ! ( ) 2 0 ln 1 SPT w g h - = =- - !
SPT type theory is to add a term proportional to into eq. (3.13). However, the immediate difficulty one meets here is that the divergence of this term at makes the usual matching procedure impossible at this point. One possible way to circumvent this difficulty is to replace by with being a positive constant to be determined later, and n can be considered form one to four. This additional term will change the formulation of other terms, (i = 0, 1, 2) and P. And the influence will be decreased with the increasing of n. If n is equal to one, all results of (i = 0, 1, 2) and P will be modified. If n is equal to four, only results of P will be changed.

Here, only the version of n = 3, which has the best prediction of non-Hadwiger term, will be introduced here, the detail results of other versions will be presented in Appendix A. The convergence factor removes the singularity at and keeps the correct asymptotic form of the non Hadwiger term for large values of . So, we propose the following expression for the chemical potential to insert a scaled particle with a radius of ( ),

. (3.25) To determine the parameters, ( ), we following a procedure similar to that used in SPT, i.e., matching eq.( 3.24) with eq.( 3.1) at as follows, ,

, (3.27) , (3.28) , (3.29) where

1 s R 0 s R = 1 s R ( ) 3 1 s R s e R d - - d i w i w 0 s R = s R s R 0 s R > ( ) 3 / 3 3 2 1 0 1 2 1 ( , ) 1 4 1 ( ) ln 2 3 s R s s s s s s e N R R R P R R R V R d p b b f ff f - - - D + ae ö F = L + + + + + ç ÷ è ø n f 1,0,1, 2 n = - 0 s R + = ( ) ( ) 0 0 s s s R R bµ b + + = = F = ( ) ( ) 0 0 s s s R R bµ b + + ¢ ¢ = = F = ( ) ( ) 0 0 s s s R R bµ b + + ¢¢ ¢¢ = = F = ( ) ( ) 0 0 s s s R R bµ b + + ¢¢¢ ¢¢¢ = = F = , ( 3.30)
, (3.31) , (3.32) and the third derivatives is equal to, (3.33) where is the compressibility. From this matching, we obtain, ,

0 ln 1 s s R bµ h + = =- - ( ) 1 3 0 1 s s R R h bµ h + ¢ = = - ( ) ( ) ( ) 2 2 3 2 1 0 1 s s R R h h bµ h + + ¢¢ = = - ( ) ( ) ( ) ( ) ( ) 2 3 3 3 6 1 7 9 1 0 1 1 s s Z R R R h h h h bµ h h + + + - ¢¢¢ = = - - - Z P b r = ( ) 0 0 ln 1 w f h = =- - 1 1 1 3 1 w R h f h = = - 3 3 1 1 2 2 2 2 2 2 1 1 3 3 2 1 1 R R w R R R R f f h h f d h h d - - é ù ae ö ae ö ae ö = - = + - ê ú ç ÷ ç ÷ ç ÷ - - è ø è ø è ø ê ú ë û ( ) ( ) ( ) ( ) 2 3 4 1 3 5 8 5 5 2 3 1 3 1 P R R h h h h h h f d b h r h - é ù + + - ae ö ê ú = - ç ÷ - è ø ê ú - ë û ( ) ( ) ( ) ( ) 2 3 3 1 6 9 ln ln 1 1 2 1 1 s R P R R R e R d h h b bµ b r h h h h r f d - - ae ö = F = = L - -+ + + ç ÷ - - è ø é ù ae ö + - - ê ú ç ÷ è ø ê ú ë û µ µ P µ
the iteration by substituting the SPT result for the pressure into the right-hand side (RHS) of eqs. (3.37) and (3.38), which leads to the following results after the first iteration and we name it as ASPT1, ,

The surface tension near a spherical hard wall is given by , (3.43) where , (3.44) , (3.45) , (3.46) , (3.47)

3 4 1 3 2 1 1 R R f d h h h h - ae ö ae ö ae ö = + ç ÷ ç ÷ ç ÷ - - è ø è ø è ø ( ) 3 1 3 2 1 1 ASPT SPT a h h bµ bµ d h h ae ö ae ö = - + ç ÷ ç ÷ - - è ø è ø ( ) ( ) 3 3 1 1 R a e R R d d d d - é ù ae ö ae ö = -- ê ú ç ÷ ç ÷ è ø è ø ê ú ë û ( ) ( ) 3 2 1 1 1 1 6 1 2 1 1 3 1 3 1 1 1 ln 1 ASPT SPT P P a b b h h h h d r r h h h h h h ì é ù ae ö ae ö ï = - + + - ê ú í ç ÷ ç ÷ - - - - ê ú è ø è ø ï ë û î ü -- -ý þ ( ) ( ) ( ) 2 1 2 1 1 1 1 0 1 2 3 3 1 3 4 1 s ASPT ASPT ASPT ASPT ASPT s s s s R ASPT s R R R R R R R R e R d g p b g g g g g - - - - ae ö = = + + ç ÷ è ø ae ö + - ç ÷ è ø ! ! ! ! ! 3 2 1 1 2 0 0 3 2 ASPT SPT ASPT R R f g g g d - ae ö = = - ç ÷ è ø ! ! ! 1 1 1 1 ASPT SPT R g f g - - = = ! ! 1 2 0 2 ASPT SPT g f g - - = = ! ! 3 4 1 1 3 3 2 1 1 ASPT R R f h h d g h h - - ae ö ae ö ae ö = = + ç ÷ ç ÷ç ÷ - -è ø è ø è ø !
In the results given above, there remains a parameter, i.e., , to be determined. We will propose, in Section 3.3, a pragmatic recipe for this and discuss in more details the accuracy of ASPT1.

The same idea can be used to treat the multi-components system. As presented in previous work, 193 the excess chemical potential for inserting a scaled particle with a radius, , into a n-component system is (3.48) where is the probability of finding a spherical cavity with radius equal to , and the exact result for this insertion probability of a small scaled particle is ,

where is the packing fraction of species . Then the chemical potential and its first three derivatives when can be derived as following, (3.50)

(3.51) (3.52) (3.53)
where is the compressibility, and . Then the expression of the chemical potential to insert a scaled particle with a radius ( )

is same with the one for one-component system, i.e., eq. (3.25). To determine the

d s R ( ) ( ) 0 ln , ex s s s R p R µ = -é ù ë û ( ) 0 s p R { } 1 min , , s n R R R + ! ( ) ( ) { } 3 0 1 1 1 1 , min , , 0 n s s n s p R R R R R R l l l h = = - + - £ £ å ! l h l 0 s R + = ( ) ( ) 3 0 ln 1 , ex s s R bµ x + = =- - ( ) 2 3 6 0 , 1 s s R x bµ x + ¢ = = - ( ) ( ) 2 1 2 2 3 3 0 24 36 , 1 1
s s R x x bµ x x + ¢¢ = = + - - ( ) ( ) ( ) 3 0 0 1 2 2 2 2 3 3 3 3 (Z 1) 0 48 9 9 72 . 1 1 1 1 s s R x x x x x bµ x x x x + - ¢¢¢ = = + - - - - - - 1 n i i Z P b r = = å ( ) 1 2 6 n i i j j j R p x r = = å s R 0 s R > parameters,
, the following four conditions are used, ,

, (3.55) , (3.56) .

(3.57)

Then the four unknown parameters, ( ), are obtained as following, ,

.

(3.61)

By using the same iterative procedure, the chemical potential of i-component equals to , (3.62) where is the compressibility of original SPT, then by using Gibbs-Duhem equation, the compressibility of ASPT1 for multi-component system can be expressed as . (3.63) Here, I would like to mention that if the number of components is equal to one, all above results will be reduced to the corresponding ones of a one-component system, automatically.

B. Double-point matching formulation

ASPT1 presented above is not an entirely self-contained theory in the sense that it

n f ( ) ( ) 0 0 s s s R R bµ b + + = = F = ( ) ( ) 0 0 s s s R R bµ b + + ¢ ¢ = = F = ( ) ( ) 0 0 s s s R R bµ b + + ¢¢ ¢¢ = = F = ( ) ( ) 0 0 s s s R R bµ b + + ¢¢¢ ¢¢¢ = = F = n f 1, 0, 1, 2 n = - ( ) ( ) 3 4 4 0 3 2 1 2 1 0 3 2 3 3 3 3 40 40 48 48 8 3 1 3 1 1 1 Z x x x xx f d d x x x x x - é ù ae ö = - + + - + ê ú ç ÷ - - - - ê ú è ø ë û ( ) 2 1 2 1 2 2 3 3 3 2 12 2 3 1 1 x x f f x d x - é ù = + - ê ú - - ê ú ë û 2 1 3 6 1 x f x = - ( ) 0 3 ln 1 f x = - - ( ) ( ) 3 2 3 2 3 1 0 1 1 ln 1 8 2 i R ex SPT i i i i i e R R Z R R d f bµ x f x f - - - = - - + + + + SPT Z ( ) 3 0 1 2 2 0 2 3 3 3 3 3 1 1 1 Z x x x x x x x x é ù = + + ê ú - - - ê ú ë û
contains an adjustable parameter, i.e., . In the followings, we will show it is also possible to develop an augmented scaled particle theory without any adjustable parameter. To circumvent the difficulty due to the divergence of the leading non Hadwiger term, , at , we propose to extrapolate the exact result for a small scaled particle, i.e., eq.(3.12), to a thermodynamics expression for a large scaled particle in two steps instead of one used for SPT or ASPT1. The first extrapolation is carried out for a scaled particle with a radius in the region of (note that is the radius of a sphere which can contain at most three hard spheres of radius, ). We propose the following expression for the chemical potential of the scaled particle with a radius in this region, (3.64) Since the non Hadwiger term, , is not included here, there is no singularity at , we can make the matching of eqs. For a scaled particle with a radius larger than , we use the following expression for its chemical potential including the leading non Hadwiger term, ,

d 1 s R 0 s R = ( ) 0 2/ 3 1 s R d R £ £ = - R d + R 3 3 2 1 0 1 2 4 1 ( ) ln 0 2 3 s s s s s R P U R u u R u R R d V p b b ae ö L = + + + + £ £ ç ÷ è ø 1 s R 0 s R = ( ) ( ) 0 0 s s s R U R bµ b + + = = = ( ) ( ) 0 0 s s s R U R bµ b + + ¢ ¢ = = = ( ) ( ) 0 0 , s s s R U R bµ b + + ¢¢ ¢¢ = = = ( ) ( ) 0 0 . s s s R U R bµ b + + ¢¢¢ ¢¢¢ = = = ( ) 0 0 1 ln 1 2 u w h = = - - 1 1 3 1 , 1 u w R h h = = - 2 2 2 3 3 1 2 1 1 u w R h h h h ae ö = = + ç ÷ - - è ø , ( 3
The coefficients in eq.(3.75) ( , ) are determined by matching eqs.

(3.64) and (3.75) at , i.e., (3.76) ( )

( ) 2 3 2 1 3 2 1 3 0 6 9 1 8 1 9 1 1 1 6 1 6 9 3 1 1 9 1 1 1 1 2 s R F R P R P R h h h pb h h h h b h h h h h h r + é ù ¢¢¢ = ae ö ê ú = + + - ç ÷ - - - - ê ú è ø ë û é ù ae ö ae ö = + + - - ê ú ç ÷ ç ÷ - - - ê ú è ø è ø ë û ( ) ( ) 3 2 2 1 2 3 18 ( 0 ) 16 2 2 1 s P F R R g R R b h p r r + + ae ö ¢¢¢ = = = - ç ÷ è ø
( )

2 g R + 1 1 18 1 1 . 1 2 5 5 1 1 P b h h r h h h é ù ae ö = + + ê ú ç ÷ - - - è ø ë û ( ) 0 2/ 3 1 s R d R £ £ = - d ( ) ( ) ( ) 3 1 0 1 3 2 2 ( , ) 1 1 ( ) ln 4 1 2 3 s s s s s s s N R R W R w w w R d V R R d R d P w R d b p b - D + ae ö = L + + + - ç ÷ è ø ³ - + - + n w 1, 0,1, 2, n = - s R d = ( ) ( ), s s W R d U R
( ) ( ), s s W R d U R d b b ¢ ¢ = = = ( ) ( ), s s W R d U R d b b ¢¢ ¢¢ = = = ( ) ( ). s s W R d U R d b b ¢¢¢ ¢¢¢ = = = 2 3 2 2 1 1 0 0 1 , 2 R u P w d d d R w u Ru R R R R d b h r - ae ö ae ö = + + + - ç ÷ ç ÷ è ø è ø 2 2 2 1 1 1 1 2 3 P w d d R Rw Ru R u R R R d b h r - ae ö ae ö = + + + ç ÷ ç ÷ è ø è ø 3 2 2 1 1 2 2 2 6 , P w d R R w R u R R d b h r -ae ö = + - ç ÷ è ø ( ) 4 1 1 P P w d R R b h r - - ae ö = ç ÷ è ø ( ) ( ) ( ) 2 3 4 1 ( ) 6 9 ln ln 1 1 2 1 1 1 s W R R P P P d R bµ b b h h r h h r h h b h r = = ae ö = L + - -+ + ç ÷ - - è ø é ù - ae ö --- ê ú ç ÷ è ø ê ú ë û 2 3 2 25 2 5 5 3 3 1 2 1 2 5 1 1 ASPT SPT A h h h h bµ bµ h h h h é ù ae ö ae ö = - - - + ê ú ç ÷ ç ÷ - - - - ê ú è ø è ø ë û (3.85)
, (3.86) where .

(3.87)

From eq.( 3.75), we obtain the following ASPT2 result for the surface tension (dimensionless one) of an HS fluid against a spherical hard wall, ,

, (3.90) , (3.91) .

(3.92)

The same idea can be used to treat the multi-component system. The expression of chemical potential for a multi-component system can be expressed as following, ( ) Same with the APST1 for multi-component system, the above results also can be reduced to the corresponding ones of a one-component system. Also, same with the original SPT, the above results can be obtained by considering a simple mixing rule from the one-component system. 193

( ) 2 3 2 1 9 20 1 ln 1 3 2 1 1 25 5 2 5 1 ln 1 2 5 2 2 1 2 5 1 ASPT SPT P P A b b h h h r r h h h h h h h h h ì é ù ae ö ae ö = - + - - + í ç ÷ ç ÷ ê ú - - ë û è ø è ø î ü é ù ï - + - + - ý ê ú - -ï ë û þ 256 3 439 9 A - = ( ) ( ) 2 2 2 2 3 2 2 2 2 0 1 2 3 4 ASPT ASPT s s ASPT ASPT ASPT ASPT s s s R R R R R R R R R g p b g g g g g - - - = ae ö ae ö = + + + ç ÷ ç ÷ è ø è ø ! ! ! ! ! 3 2 2 0 0 3 3 4 2 3 3 ASPT SPT ASPT g g g - ae ö = - ç ÷ - è ø ! ! ! 2 2 2 1 1 3 3 6 2 3 3 ASPT SPT ASPT g g g - - - ae ö = + ç ÷ - è ø ! ! ! 2 2 2 2 3 3 4 2 3 3 ASPT SPT ASPT g g g - - - = - - ! ! ! 4 2 2 3 3 2 5 2 5 23 3 5 3 1 1 1 2 1 2 5 3 ASPT h h h h g h h h h - ì ü é ù ae ö ae ö - ï ï = - + - ê ú ç ÷ í ý ç ÷ ç ÷ - - - - ê ú è ø è ø ï ï ë û î þ ! , ( 3
( ) ( ) ( ) ( ) ( ) 3 1 3 2 1 0 1 2 2 1 0 1 2 3 ln 1 1 , 0 4 1 , 0 2 3 1 1 2 4 , 3 n s i s ex s s s s s s s s s s s R R R R R P R u uR uR R d w w w R d w R d R R d R d P l l l h p b bµ p b = - ì ï ï é ù ï - + -£ < ê ú ï ë û ï ï = + + + £ £ í ï ï + + -+ - ï ï ³ ï - ï + î å ( ) 2 3 1 i d R = - i R n w n w ( )( ) ( )( ) 3 4 4 0 1 2 2 1 0 2 3 3 3 3 3 8 144 5 144 5 8 1 2 5 1 2 5 1 1 2 5 1 SPT i i w BR Z BR x x x x x x x x x x - é ù = - + ++ ê ú - - - - - ê ú ë û ( ) ( ) 1 0 3 ln 1 4 3 2 3 i w w R x - = - -- + ( ) 2 1 1 2 3 6 18 7 4 3 , 1 i w w R x x - = + + - ( ) ( ) 2 1 2 1 2 2 3 3 3 2 3 12 24 45 26 3 , 1 1 i w w R x x x x - é ù = + - + ê ú - - ê ú ë û ( ) 97 56 3 9 B = - ( ) 2 3 2 1 3 1 0 ln 1 8 2 ex SPT i i i i i w w w R R Z R R bµ x x - = - - + + + +

Results and discussion

In this section, we are going to assess the accuracy of our two ASPT.

A. Determination of the adjustable parameter in ASPT1

Before ASPT1 can be used in practice, we still need to determine parameter, .

An appealing recipe for this is to adjust in such a way that the pressure given by eq.( 3

P P = 0.5 h = d ( ) ( ) 3 3 1 1 1 4 3ln 2 1 R e R R d d d - é ù ae ö - - = ê ú ç ÷ - è ø ê ú ë û 3 R 1 ASPT P CS P 0.234499 R d =

B. Pressure

ö - - = + - ê ú ç ÷ ç ÷ - è ø è ø ê ú ë û 0.181952 R d = d 0.234499 R d = 0.234499 R d = 0.181952 R d = P b r
, significantly improve the accuracy of pressure. This can be considered as evidence of morphological thermodynamic is only a good approximation. Fig. 3.2 also shows ASPT2 slightly overestimated the pressure at high packing fraction. In Table 3.1 we compare the first 10 Virial coefficients obtained from ASPT1 and ASPT2 with the prediction from original SPT, Carnahan-Starling equation of state, and the exact values. One unanticipated finding is that the prediction of Virial coefficients from ASPT2 is more accurate than all other versions, while ASPT2 was found to slightly overestimate the pressure in Fig. 3.1. From Table 3.1, it shows that all equations of state can offer exact first three Virial coefficients, all equations of state overestimate the Virial coefficients but the ASPT1 underestimate it. The choice of the parameter, , is the reason, and if we decrease , the prediction will be closer to the exact ones,

C. Surface tension

Now, we proceed to assess the accuracy of ASPT1 and ASPT2. SPT does not only provide the expression of some thermodynamic properties but also provides the expressions of some surface properties, such as surface tension and Tolman length.

2 0 1 2 s s R R R R g g g g - - ae ö = + + ç ÷ è ø ! ! ! ! s R 1 0 T R g d g -= ! ! 1 0 T R g d g -= ! ! 2 4 g p - ! 1 g - ! 2 g - ! 1 g - ! 1 g - !
lower than simulation results. In high packing fraction, the prediction of CS-SPTM is higher than the results from other equation of state. Because lack of simulation result, we do not know which result is correct in high packing fraction, it means that the future study is needed.

= + + + +O ê ú ç ÷ ç ÷ ê ú è ø ë û ! 1 g - !
with the exact result, this can explain why the prediction of from CS-SPT is bad.

The prediction from CS-SPTM is also away from the exact one, but much better than CS-SPT. The prediction from ASPT2 is closer to the exact value than original SPT. One unanticipated finding was that the prediction from MFMT is more accurate, while the prediction of is not good.

g - ! 1 g - ! 1 g - ! 2 g - ! 2 g - ! 2 g - ! 2 g - ! 2 g - ! ( ) 2 4 2 1 81 3 289 1 2 16 105 g h h h h p - é ù ae ö = + + - +O ê ú ç ÷ ç ÷ ê ú è ø ë û ! 2 g - ! 2 g - ! n Exact SPT MFMT ASPT2 2

D. Non-Hadwiger coefficient

As said at the beginning of this paper, we want to build a simple analytical equation of state that can provide an acceptable prediction of Non-Hadwiger coefficient. In this part, the non-Hadwiger obtained from different equation of state are considered.

Hansen-Goos 221 has reported a very accurate expression of Non-Hadwiger coefficient, however, his theory, SFMT, need planar surface tension and Tolman length as input.

E. Multi-component system

Now, the accuracy of the two ASPT for the multi-component system is considered.

The comparison of the prediction of excess chemical potential from our two new ASPT, original SPT, and BMCSL equation of state 260 are shown in Fig. 3.6. It shows that ASPT1 and ASPT2 strongly improve the prediction from original SPT for both big particle component and small particle component, and the results are nearly indistinguishable from the BMCSL equation of state on the scale of the plot. In original SPT, if the number of components is reduced to one, all results will reduce to the corresponding ones of a one-component system. It can be readily checked that ASPT1 and ASPT2 also have this property. Also, ASPT1 and ASPT2 can be obtained by using some simple mixing rules, just like showed in 193 . All results above show our two new methods are reasonable and consistent. The BMSL results are presented by the green line, the black line is ASPT1, the red line ASPT2, and the blue line original SPT.

Concluding remarks

In the present work, we found the original expression of the chemical potential is not completed. Based on previous works on SPT and morphological thermodynamic, we reported two new ways to present the chemical potential. The first one contains an arbitrary parameter. The second one uses two Laurent series to describe the chemical potential. Both of them do not only significantly improve the accuracy of thermodynamic properties, i.e. pressure and chemical potential, but also improve the accuracy of surface properties, i.e. surface tension and Tolman length. Also, the same idea can be used to treat the multi-components hard sphere fluids system, and the improvement is significant.

The success of ASPT1 and ASPT2 indicates that the morphological thermodynamic is not an exact theory, it is only a very good approximation. However, ASPT1 can only provide an acceptable result of Non-Hadwiger coefficient, which means we still need future study in this term. Since the original expression of the reversible work is not completed and when we add an additional term, , the accuracy has significantly improved. It is natural to ask a question if the more terms, and , are added, the higher accuracy of SPT will be obtained?

Unfortunately, since lack of the understanding of reversible work of inserting a scaled particle, eq.( 3.4), we do not have enough conditions to build a new SPT that contains more term.

1 R

2 1 R 3 1 R

A general equation of state for confined fluid: morphological thermodynamics 4.1 Introduction

In theoretical studies, for describing the porous material, many models are introduced, i.e., hard sphere matrix model, overlapping hard sphere matrix, hard sponge matrix, and Slit pore, as shown in Fig. 4.1. The models of disordered porous medium are always built with the help of some fixed particles. Although all three models for random porous medium introduced above has analytical equation of state obtained from SPT, the derivation of all those equations of state is very complex and depends on the model of porous medium. So, it still remains a great challenge to build a general framework to understand the behavior of fluid confined in different porous medium.

Although all three models for random porous medium introduced above has analytical equation of state obtained from SPT, the derivation of all those equations of state is very complex and depends on the model of porous medium. So, it still remains a great challenge to build a general framework to understand the behavior of fluid confined in different porous medium.

In recent, Mecke and his co-workers proposed a general framework, named as morphological thermodynamics. 207,209,210,234 This framework gives us some hope to build a general theory for different confined fluid. This method is from a beautiful mathematics theorem, Hadwiger's volume theorem. 211,212 It is not only a beautiful mathematics theorem but also offers a very powerful way to study confined fluid.

γ terms may be equal to zero. In fact, their studies considered a fluid near spherical or cylindrical particle, it means that the interface between fluid and the obstacle particle is concave from fluid view, while the Hadwiger's volume theorem is only valid in convex interface.

The remaining Chapter is organized as follows. A brief introduction of morphological thermodynamics and a general equation of state for different confined fluid will be presented in the next section. Then the accuracy of the general equation of state in different confinement systems will be assessed in Section 4.3.

Theory

A. A brief introduction of Morphological thermodynamics

Hadwiger's volume theorem is one of the most important and beautiful theorems in geometric convexity.

: d k µ ® R d k R 0 ( ) ( ) d i i i cV µ = = å K K i c 3 V 2 V 1 V 0 V W 0 1 2 pV A C X f s k k W = - + + + V 0 f A C X p s 1 k 2 k ε ∞ < ⎧ ⎪ = ≤ < + ⎨ ⎪ ≤ + ⎩ of the coordinate.
is the radius of the matrix particle. In this work, is used to denote matrix component, is the well depth and is the width of the well. Then the grand partition function is , (4.4) where is the activity, is the chemical potential. The grand potential is equal to (4.5) The integration in the equation above, , can be derived as

= - + - + + ç ÷ ç ÷ è ø è ø 3 0 0 4 1 3 R V f p = - 2 0 4 A R p = 0 4 C R p = 4 X p = 0 1 2 zV A C X b f s k k W = - + + + ( 1 ) z e d be s - = - - 2 1 ( 1) z e d be k - = - - 3 2 ( 1) 3 z e d be k - = - -
spherical container, ideal gas fluid around or confined in a cylinder, the same results can be also obtained. Since the details of the calculation are similar to this example, we will not show them here. We also found in some cases this expression is not established, such as, when fluid around two big spheres and there is a region that fluid suffered interaction from both two big spheres. (see Appendix B). It should be mentioned that, is equal to the pressure of the equilibrated bulk system. Maybe those four coefficients can be obtained from the equilibrated bulk system, instead of the bulk system has the same fluid density as in Chapter 2. One should note that, normally, the surface area of the matrix particle has two different definitions 220 , the first is equal to which called solid surface area, the second is which called reference surface area, where is the radius of fluid particle. When the fluid is ideal gas fluid, those two definitions are equivalent. In the following work, we will use the first definition. Hadwiger's volume theorem is about a mapping from a convex set to a number, however, in above example, the interface between fluid and matrix particle is concave from the fluid's view, and in Hansen-Goos' work 221 , he also considered a fluid around a spherical particle. This situation does not satisfy the Hadwiger's volume theorem, so we also calculated the fluid confined in a spherical container, we found the non-Hadwiger term still exists. We have to say the morphological thermodynamic is only a very accurate approximation.

B. A new general equation of state

In this section, a new general equation of state for a hard sphere fluid confined in different porous medium will be introduced. In morphological thermodynamics expression, the grand potential of a confined fluid is equal to , (4.15) here, the independent variables are chemical potential, , and temperature, T, which means we assume those four coefficients can be obtained from the equilibrated bulk system. Compare the grand potential, we are more interested in adsorption amount, with

z 2 0 4 R p 2 0 1 4 ( ) R R p + 1 R 0 1 2 ( , ) ( , ) ( , ) ( , ) p T V T A T C T X b µ f s µ k µ k µ W = - + + +
µ the help of thermodynamic relation, (4.16) the relation between chemical potential and confined fluid density, , is . (4.17) Since the relation between the chemical potential and fluid density for bulk system is already known. With the help of chain rule, the equation above can be rearranged as (4.18) The well-known Gibbs-Duhem relation is . (4.19) The relation between confined fluid density and bulk fluid density is equal to . (4.20) The relation between those three coefficients and bulk fluid density can be obtained from scaled particle theory 193 . where is the packing fraction of bulk fluid, and is the fluid in previous works [START_REF] Holovko | Fluids in Random Porous Media: Scaled Particle Theory[END_REF] . The derivation of our new general equation of state is easier to scaled particle theory, while the final expression is more complex than scaled particle theory.

, N b bµ ¶ W = - ¶ c r 1 2 0 ( , ) ( , ) ( , ) ( , ) c T T p T T A C X V V V k µ k µ µ s µ r f µ µ µ µ ¶ ¶ ¶ ¶ = - - - ¶ ¶ ¶ ¶ 0 1 2 ( , ) ( , ) ( , ) ( , ) b b b b c b b b b b b b b p T T A V T T C X V V r r s r r r f r µ r µ k r r k r r r µ r µ ¶ ¶ ¶ ¶ = - ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ - - ¶ ¶ ¶ ¶ b b b T T p µ r r r ae ö ae ö ¶ ¶ = ç ÷ ç ÷ ¶ ¶ è ø è ø 1 2 0 ( , ) ( , ) ( , ) b b b b b b c b b b b T T T A C X V V V s r r k r r k r r r r f r µ r µ r µ ¶ ¶ ¶ ¶ ¶ ¶ = - - - ¶ ¶ ¶ ¶ ¶ ¶ 2 2 1 3 (2 ) 1 , 4 2(1 ) b b b R h h bs p h + = - 1 1 3 1 4 1 b b R h bk p h = - 2 1 ln(1 ) 4 b bk h p = - - 2 3 3 3 14 13 5 ln( ) ln(1 ) 2(1 ) b b b b b b h h h bµ r h h - + = L - -+ -

Results and discussion

In this section, we are going to assess the accuracy of our new general equation of state in the different porous medium, i.e., Madden-Glandt model, sponge-like models, slit pore model. Our new results will be compared with molecular simulation and scaled particle theory [START_REF] Holovko | Fluids in Random Porous Media: Scaled Particle Theory[END_REF] . We will start from the disordered porous medium.

A. Madden Glandt disorder porous medium

Here, we consider two kinds of Madden-Glandt models, the hard sphere one (HS) which the interaction between matrix particles are hard sphere potential and overlapping hard sphere one (OHS) which has no interaction between matrix particles. For the HS matrix, the expressions for four geometric properties are

B. Hard sponge disordered porous medium

In this section, our new equation of state will be applied to describe hard sphere fluid confined in a hard sponge matrix. This model can be considered as an opposite system of the OHS matrix. Not like OHS matrix, in this model, the material particles are considered as cavity. It should be noted that the interface between the matrix and fluid is a convex surface in this model, while in OHS matrix, it is a concave surface.

The

D. Slit pore system

All models we considered in the previous section have a curved surface, a simple porous medium model, slit pore, which has a planar interface between fluid and porous medium, will be considered this section. In this model, the porosity, , is equal to one, and the surface area per unit volume equals , (4.37) where is the distance between two solid surfaces. Two curvatures, and , are equal to zero. Since this model is simpler than previous models, we consider if we can use a simpler equation to describe this system. Here, the fluid density and temperature are considered as independent variables. Then from morphological thermodynamics, the free energy, , of a hard sphere fluid confined in a slit pore can be given by: , (4.38) where is the free energy of bulk system which has the same fluid density with confined system, and is the surface tension of a hard sphere fluid near a planar wall, since there are two planar walls in this system, the surface area of solid-fluid interface is . and can be offered by SPT:

, (4.39) (4.40)
where is the packing fraction, and is the radius of fluid particle. Then the chemical potential of fluid confined in slit pore can be obtained as following, (4.41) where is the distance between two planar walls.

0 f / 2 / A V H = H C X s F b 2 s b F F A b b bg = + b F b bg 2A b F b A bg 2 2 3 1 1 1 6 8 R b h g h p h h ae ö + ç ÷ - é ù = ê - ë è ê ø ú ú û 2 3 3 ln( 1) ln (1 ) 3 1 2 , 1 b F V b h h r r r h h h é ù ae ö ê ú ç ÷ - è ø ae ö = L ê ú ë -+ - -+ + ç ÷ û - è ø h R / 2 / 2 , b s b F V A V L b bg bg bµ bµ r r r ¶ ¶ ¶ = + = + ¶ ¶ ¶ L (4.42) (4.43)
where is the diameter of fluid particle. Here, we obtain an equation of state for a hard sphere fluid confined in a hard wall system. However, in this system, we are more interested in a fluid confined in a square well wall system. To determine this equation of state, we use first order perturbation theory to calculate the equation of state for this system. The interaction between fluid and wall, , is

, (4.44)
where is the width of fluid-wall potential, is the depth of fluid-well potential.

Here, the reference system is the system that which is the hard wall system.

Hence, the interaction between fluid and wall can be divided into two parts, the reference part, and the perturbation part, ,

Then the free energy is , (

where is the free energy of a hard sphere fluid confined in a hard wall system, , and is the ensemble average of reference system, which means the perturbation part will not change the fluid density profile. The ensemble average can be rewritten as

3 2 3 3 1 6(1 2 ) 4 1 2 , 8 (1 ) 3
(1 ) R R R bg bg h h h p r h r p h h ¶ ¶ ¶ + + = = = ¶ ¶ ¶ - - 2 3 3 3 3 14 13 5 1 2 ln( ) ln(1 ) , 2 (1 )
(1 )

s L h h h s h bµ r h h h - + + = L - -+ + - - s 10 u 10 , | | / 2 ( ) , / 2 | | / 2 0, | | / 2 h L u h L d h L h L d e ¥ > ì ï = -£ < í ï < - î d e 0 e = 10 10 10 ( , ) ( ) (
)

ref per u h u h u h l l = + 10 0, | | / 2 ( ) , / 2 | | / 2 0, | | / 2 per h L u h L d h L h L d e > ì ï = -£ < í ï < - î 01 0 sw s F F w b b b = + á ñ s F 01 01 1 ( ) N per i i w u h = = å 0 á ñ ! . (4.48)
Now, once we know the density profile of the reference system, the free energy can be obtained. We assume the excess adsorption is adsorbed in the region where the interaction between solid surface and fluid is equation to , and in the remaining region, the fluid density is equal to the fluid density of the equilibrated bulk system, . The density distribution is (4.49) For the reference system, the bulk density can be solved from the Gibbs adsorption equation, ,

where is the excess adsorption amount per unit area at the interface. It can be obtained as

, ( 4.51) , (4.52) , (4.53)
where is the average density. Then the ensemble average of is equal to

. ( 4.54) Since, . (4.55)
As we known,

01 0 01 ( ) ( ) ref per w A h u h dh r á ñ = ò e b r 0, | | / 2 ( ) , / 2 | | / 2 , | | / 2 ref b c h L h L d h L h L d r r r > ì ï = -£ < í ï < - î N T bg bµ ae ö ¶ = G ç ÷ ¶ è ø N G 1 ( ) ( 2 2 ) ( ) 2 2 N b c b c b A h dh d d d A r r r r r r G = - = ´-´= - ò 2 N b L r r G = - 1 2 c N d L r r ae ö = + - G ç ÷ è ø r 01 w 01 0 1 2 2 2 c N w A d dA d L r be be r é ù ae ö á ñ = = + -G ç ÷ ê ú è ø ë û N T T T bg bg h bµ h bµ ae ö ae ö ae ö ¶ ¶ ¶ G = = ç ÷ ç ÷ ç ÷ ¶ ¶ ¶ è ø è ø è ø , ( 4.56)
. (4.57) Hence, the excess adsorption amount is (4.58)

The dimensionless excess adsorption amount is

. (4.59)
Then the free energy of this system can be obtained, ,

and the chemical potential is . (4.61) From the equation above, one can find that the first term is the bulk chemical potential, the second term denotes the interaction between a hard sphere fluid and a hard wall, and the last term denotes the interaction between a hard sphere fluid and the square well interaction. To verify this eq. ( 4.61), the molecular simulation for a hard sphere fluid confined in slit pore were performed. In Fig. 4.6, one can find eq. ( 4.61) has great accuracy in different conditions, even in a high fluid density. To our best knowledge, for this moment the results presented in our study provide the most accurate theoretical description of the thermodynamic properties of a hard sphere fluid confined in a slit pore.

2 (1 ) (1 2 ) (5 4 ) / L h h h bµ h sh h ¶ - = ¶ + + + 3 3 3 1 2 4 (1 ) R bg h h p h ¶ + = ¶ - 3 2 3 ( 1 2 )( 1 ) . 4 (1 2 ) (5 4 )/ N R L h h h p h sh h + - G = + + + 3 * 4 3 N N R p G = G 01 0 1 2 1 2 sw s s N F F F d w V V V V L d L b b b be r é ù ae ö = + á ñ = + + -G ç ÷ ê ú è ø ë û * 3 / 1 2 2 1 2 1 2 (1 ) sw N sw b F V d R L L d L b s h be bµ bµ r h h é ù ae ö ¶ ¶ G + = = + + + - ê ú ç ÷ ¶ - ¶ è ø ë û µ indistinguishable from

Quench effect on the chemical potential

Introduction

In 1988, Madden and Glandt proposed an interesting model for random porous material and developed a statistical-mechanics for investigating the structure and thermodynamics of fluids confined in random porous media. [START_REF] Madden | Distribution Functions for Fluids in Random Media[END_REF] In this model, the disordered porous material is mimicked by a random matrix that can be generated from quenching a fluid system at its equilibrium state. Thus, the quenched "fluid" particles constitute the matrix (thereafter called matrix particles), and the voids among them are pores. This simple model successfully characterizes the main features of disordered porous materials, e.g., pore connectivity, pore size distribution, etc., and thus have been widely applied in both molecular simulations and theoretical studies of fluid confined in random porous media. Since the fluid species are mobile within the matrix configuration, this model is also called as quench-annealed mixture with quenched species referring to the matrix and annealed species referring to the confined fluid. It is to point out also that the way for constructing such a porous matrix mimics the spinodaldecomposition procedure for fabricating some porous materials experimentally.

Although many investigations have been devoted to study fluids confined in random porous media since Madden-Glandt model was proposed, no precise measure has ever been proposed to describe quantitatively the quench effect for the adsorption of a fluid in a random porous material. We propose to measure such an effect by the difference of free energy to insert a fluid particle into a quench-annealed system or into the corresponding equilibrium binary mixture. It can appear perplexing that there are still quite some fundamental questions related to quench effect that we cannot answer straightforwardly. Without attempting to be exhaustive, we can list the following ones:

1) What is the consequence of quenching?

2) Does a quench-annealed system or its corresponding equilibrium binary mixture appear more crowded? Crowdedness means here more difficult to insert another fluid particle, i.e., with high chemical potential.

3) When the quench effect becomes significant? 4) In the case of a large quench effect, which are the dominant factors contributing to the considerable increase of chemical potential?

In this chapter, we will address these issues by studying some simple model systems.

In the next section, we present first the considered model and the methods we used for carrying out our investigation. In Section 5.3, the obtained results are presented, analyzed and discussed. The insights gained from this study are summarized in the last section and we point out also the implications useful for designing new functionalized porous materials, e.g., for high-capacity gas adsorbents, etc.

Model and methods

A. Model

In the present work, we consider a hard sphere (HS) fluid (denoted as species 1) of particles confined in a quenched HS matrix (denoted as species 0) of particles. The interactions between the particles of the different species are described by, ,

where and are respectively the radius of fluid and matrix particles, and are respectively the position vectors of the ith and the jth fluid particles while and are those of matrix particles.

1 N 0 N 1 11 1 2 ( ) 0 2 i j i j i j R u R ì ¥ -< ï - = í -³ ï î r r r r r r 1 0 10 1 0 ( ) 0 i j i j i j R R u R R ì ¥ - < + ï - = í - ³ + ï î r q r q r q 0 00 0 2 ( ) 0 2 i j i j i j R u R ì ¥ - < ï - = í - ³ ï î q q q q q q 1 R 0 R i r j r i q j q

B. Theory

The fact that the matrix particles in a quench-annealed system are immobile has some consequences on both structural and thermodynamic properties. In the present work, we focus on the quench effect on thermodynamics only and examine in particular such effect on the chemical potential in order to learn how quenching affects the insertion of a fluid particle. The difference between the chemical potential for inserting a fluid particle into a quench-annealed system and that for inserting the same particle into the corresponding equilibrium binary mixture appears to be a natural measure for quantifying the quench effect, i.e., .

where (kB: Boltzmann constant; T: temperature). In the followings, we call quench chemical potential. Now, we point out first that the measure defined in eq.( 5.4) has an exact diagrammatic expansion. Following the work of Madden and

Glandt [START_REF] Madden | Distribution Functions for Fluids in Random Media[END_REF] and that of Morita and Hiroike [262][263][264] , one obtains readily the following expansions, ,

and

1 1 1 QA Equil b µ bµ bµ D = - ( ) 1/ B k T b = 1 b µ D ( ) 3
.

(5.9)

Eqs.(5.5) and ( 5.6) yield straightforwardly, ,

where circles denote fluid points and squares denote matrix points. Due to its diagrammatic characteristics described in eq.( 5.10), we call sometimes shielding chemical potential as well (an interchangeable name with quench chemical potential). We will discuss in the next section how some interesting information about quench effect can be deduced from the formal results presented here.

C. Application of scaled particle theory

It is also well-known that even for simple models like HS, it is not possible to calculate exactly and analytically the sum of graphs like that in eq. (5.10). Nevertheless, various approaches, some being even analytic, exist for calculating chemical potential in both an equilibrium binary system and a quench-annealed system. In particular, scaled particle theory (SPT) is a well-known and successful approach which allows for ( ) ) for the chemical potential of an HS fluid in an HS matrix. [START_REF] Holovko | Fluids in Random Porous Media: Scaled Particle Theory[END_REF] This is why for our study here we use SPT2b1 and its expression for the chemical potential of an HS fluid in an HS matrix is given by, (5.11) where , (i = 0, 1) is the packing fraction, is the ratio of the radius of fluid particle to matrix particle, is the geometric porosity, and is the chemical potential to insert one fluid particle into an empty matrix which is given by (5.12) and in eq. (5.11) is the matrix porosity measured with a fluid particle as the probe (note that is the porosity measured with a point particle), . (5.13) In order to facilitate comparison, we cast the expression of chemical potential of the corresponding binary mixture in a form similar to that given in eq. (5.11), i.e., ( )

i j i j u f e b - - - = - q q q q 01 ( ) 01 1 ( ) i j i j u f e b - - - = - q r q r 11 ( ) 11 1 ( ) i j i j u f e b - - - = -
( ) 2
( ) 2 0 1 1 1 0 2 2 2 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 2 1 0 1 0 ln(1 / ) 7 3 4 9 1 1 / / 15 3 10 18 1 / 2 1 1 1 1 / / 3 1 1 1 / ex QA ex I h h f f bµ bµ h f t t t h h f h f h h h f t t t h f h h h f th h f h h f é ù ae ö - ê ú = - - + + + + + ḉ ÷ - - ê ú è ø ë û é ù ae ö ae ö ê ú + + + + ç ÷ ç ÷ - - - - ê ú è ø è ø ë û ae ö ae ö + + ç ÷ ç ÷ - - è ø è ø 3 ( ) _ 3 1 1 1 1 ln ex QA QA bµ bµ r = -L 3 4 /3 i i i R h p r = 1 0 / R R t = 0 0 1 f h = - 1 ex I bµ ( ) ( ) 2 0 0 0 0 3 3 0 0 0 2 1 3 3 ln(1 ) 3 3 3 1 1 2 2 1 ex I t t h h h bµ h t t t t h h h ae ö ae ö = - - + + + + + ç ÷ ç ÷ - - - è ø è ø + f 0 f 1 ex I e bµ f - = (5.14)
The two first terms on the right-hand side of both eqs. (5.11) and (5.14) are identical.

The rest of the RHS of eqs. (5.11) and (5.14) is a third order polynomial of but with different coefficients for or . So, the quench effect shows up through these coefficients.

D. Simulation

In order to check the validity of the prediction about the quench effect given by SPT, we carried out a series of Monte Carlo (MC) simulations 226,265 of an HS fluid in an HS matrix. A cubic simulation box was used with periodic boundary conditions.

Each simulation run is performed with a particular matrix realization and the fluid particles can move in the volume which is not occupied by matrix particles. The matrix configurations are generated by using canonical ensemble Monte-Carlo simulations (CEMC). Since matrices of finite size are used in the simulation, any observable quantity fluctuates with matrix realizations and hence an average over matrix realizations should be taken also. In the present work, we generated matrix realizations of sizes ranging from 200 to a few thousands matrix particles. The average over matrix configurations is calculated with typically 20 realizations. For each matrix realization, a GCEMC simulation about 10000 trial moves per particle is performed during which the average fluid density is calculated. For all the numerical results presented below, we use the diameter of fluid particle, i.e., , as the unit of length. Matrices with different particle radii ( ) as well as of different porosities were

( ) ( ) ( ) ( ) 2 ( ) 2 2 1 1 1 0 3 3 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 3 3 1 0 1 0 ln(1 / ) 7 15 6 6 3 1 1 / 9 15 3 7 3 15 2 1 1 / 2 1 / 18 1 1 1 1 / ex Equil ex I h h bµ bµ h f t t t t t h h h h f h t t t t t h h f h h h h f t h h h f é ae ö ê = - - + + + + + + ç ÷ - - ê è ø ë ù ae ö é ú + + + + + + ḉ ÷ ê - - - ú è ø ë û ù ae ö ae ö ae ú + ç ÷ ç ÷ - - - ú è ø è ø è û 0 2 3 3 1 0 0 1 0 / 3 1 1 1 / th h f h h f ö ae ö ae ö + + ç ÷ ç ÷ ç ÷ - - ø è ø è ø 1 0 1 0 / 1 / h f h f - _ 1 ex QA bµ _ 1 ex Equil bµ 1 1 2R s = 0 1 0.5 2.5 R s = - considered.
To determine the quench effect, it is necessary to calculate the chemical potential of an equilibrium HS binary mixture corresponding to the quench-annealed system. To facilitate the comparison between the two systems, we performed CEMC simulations for the equilibrium HS binary mixture so that the densities of the two components can be prescribed. The chemical potential is calculated by using Widom's test particle method 228 .

Results and discussion

In this section, we will show successively in which cases there is no quench effect at all, when it starts to manifest itself and the situations in which the quench effect becomes strong.

A. Strictly vanishing quench effect

Although the results given in eqs. (5.6) and (5.10) are quite formal ones, they can provide already some interesting information about the quench effect. Eq. (5.10) shows immediately that there is no quench effect at the order of since all the diagrams in the sum given in eq. (5.10) are of an order equal or higher than . Moreover, if all the diagrams in the sum given in eq.( 5.6) have only one fluid point (it is necessarily the white one), none of these diagrams contains any shielding set. Hence, the quench effect vanishes in this case as well. This corresponds to the situation that there is only one particle of species 1. So, insertion of a fluid particle into an empty matrix (no fluid particle inside the matrix) or into a bulk fluid of species 0 (not quenched but mobile particles) costs exactly the same amount of energy. The above discussion shows that the quench effect appears only when some fluid-fluid and matrix-matrix interactions exist at the same time, i.e., beyond the case of a single fluid particle and beyond the linear order of matrix density.

In the case of a single particle of species 1, it is possible to calculate the excess

0 r 2 0 r ⎧ ∞ - < ⎪ - = ⎨ - ≥ ⎪ ⎩ ⎧ ∞ - < ⎪ - = ⎨ - ≥ ⎪ ⎩ ρ μ μ π β β ⎛ ⎞ = - - ⎜ ⎝ = ⎟ ⎠
ρ matrix with density ), the results given in eq.( 5.17) hold as well. It is worth to note also that in the limit of , the SPT results given in eqs. (5.11) and (5.14) reduce both to that of eq.( 5.17).

In the case of one HS of radius, , in a matrix of point particles (see Fig. 5.2 for a sketch), i.e., the interaction potential being, , (5.18) the excess chemical potential can be also calculated exactly and analytically and we obtain, . (5.19) Once again, this explicit calculation confirms the conclusion obtained from the diagrammatic expansion, i.e., no quench effect at the level of a single fluid particle.

Moreover, eq. (5.19) shows that the excess chemical potential is of the linear order of and this result confirms also another conclusion from the diagrammatic expansion, i.e., the quench effect manifesting itself at the order equal or higher than . In this case, SPT results, i.e., eqs. (5.11) and (5.14) reduce again to the exact one of eq. (5.19).

Here, one can raise naturally the question whether the SPT results given in eqs. (5.11) and (5.14) are consistent with the exact result that the quench chemical potential vanishes at the linear order of matrix density. From eqs. (5.11) and (5.14), we obtain the following expression for the quench chemical potential,

0 r 1 0 R ® 1 R 1 10 1 ( ) 0 i j i j i j R u R ì ¥ - < ï - = í - ³ ï î r q r q r q 3 ( ) ( ) 1 0 1 1 4 3 ex QA ex Equil R b p µ bµ r = = 0 r 2 0 r ( ) ( ) ( ) ( ) β μ βμ βμ φ φ η η η η φ τ τ τ τ τ τ φ η η η η φ η η η η φ τ τ τ τ τ η η η η φ Δ = ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ - ⎢ ⎥ = - + + - + - ⎜ ⎟ ⎜ ⎟ - - - - ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎢ ⎥ - + + + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ - - - - ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ - ⎠ ⎣ ⎦ - τη η η η φ τ τ η η η η φ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎢ ⎥ + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ - - - - ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ β μ Δ η ( ) ( ) η η τ τ φ φ β μ η η η τ τ τ τ η φ η η η η → → + ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ - Δ = - + + - - ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ - - - ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ( ) η φ φ τ τ τ φη → -= + +
Substituting eq.( 5.22) into eq.( 5.21), we see that the first term on the right-hand side of eq.( 5.21) vanishes. So, at the order of , the quench chemical potential is strictly zero. It is known that SPT for a bulk binary mixture is exact to this order while, to our best knowledge, no calculation of any Virial coefficient has been calculated for a confined fluid. The vanishing quench chemical potential at the order of implies that our SPT for confined fluid is also exact to this order. The non-vanishing part in eq.( 5.21) comes only from the contributions of the higher order terms for which SPT does not give exact results. So, to the orders for which SPT is exact, the result for quench chemical potential from SPT is consistent with the exact result, i.e., vanishing quench effect at the linear order of matrix density. It is quite reassuring to see, from the above discussions, that SPT either reproduces the exact results in some particular cases or is consistent with the general conclusion for vanishing quench effect.

In a more general situation, i.e., finite fluid and matrix densities, SPT results in eqs. (5.11) and (5.14) show that quench effect vanishes also in the limit to , i.e., (5.23) The vanishing quench effect in this limit of very large matrix particle cannot be deduced by a straightforward inspection of the diagrams in eq. (5.10). We have not succeeded in finding a formal proof. Therefore, we consider this prediction of SPT for vanish quench effect in case of very large matrix particles as a conjecture and this deserves certainly further investigations.

Finally, it is worth to note also that appears in eq. (5.23) only in the form of but no linear term in neither terms involving are present. This implies that in the limit of , the surface free energy between the fluid and the matrix does not contribute to the chemical potential.

B. Small quench effect

At the end of the last subsection, we have just seen that quench effect vanishes

0 1 r r 0 1 r r 0 t ® 2 3 ( ) ( ) 1 0 1 0 1 0 1 1 1 1 0 1 0 0 0 1 7 / / / 15 ln( ) 3 1 / 2 1 / 1 / ex QA ex Equil h f h f h f bµ bµ f h h f h f h f ae ö ae ö = - + + + ç ÷ ç ÷ - - - è - è ø ø = 0 R 3 0 R 0 R 2 0 R 0 t ®
when the matrix particles are much larger than the fluid particles, i.e., .

The extensive numerical calculations we have carried out show that for all the cases with matrix particles larger than fluid particles, i.e., , the quench effect remains always quite moderate. The results for are presented in , ). It is to note also that for both densities, it is slightly more difficult to insert a particle of species 1 into the equilibrium binary mixture than into a matrix at high fluid density, i.e, . We carried out also MC simulations under the same conditions and the results are plotted also in Fig. 5.3 along with the SPT ones. We see that there is a very good agreement between SPT and MC results for both quench-annealed and equilibrium binary systems.

1 0 / 0 R R t = ® 0 1 R R > 0.5 t = 0 1 t < < t * 0 0 0.5 r < < * 1 0 0.5 r < < 1 1
Equil QA µ µ >

C. Turning point for the change from small to large quench effects

The results for the case that the size of fluid and matrix particles is the same, i.e., , are presented in Fig. 5.4. When the matrix density is relatively low ( ), the quench effect remains small over the whole range of the considered fluid densities.

Nevertheless, it is to note that the chemical potential of the quench-annealed system is higher than that of the corresponding equilibrium binary mixture, i.e.,

(opposite to what is found for ). The new feature shown by Fig. 5.4 is that at , the quench effect becomes significant when the matrix density is higher In Fig. 5.6, the results for and two matrix densities ( : 0.0384 and 0.05) In order to understand the large quench effect found here, we consider the case of a matrix with point particles, i.e., . Noting that in the limit ( ), only survive for the terms involving in eqs. (5.11), (5.12), and (5.14), we

QA Equil µ µ > 1 t > 1 t = 1 1 Equil QA µ µ > 1 1 QA Equil µ µ > * 3 1 1 1 (2 ) R r r = 1 0 / 1 R R t = =
* 3 1 1 1 (2 ) R r r = 1 0 / 2 R R t = = ( )
* 3 1 1 1 (2 ) R r r = 1 0 / 5 R R t = = ( ) 3 0 0 * 0 2 0.0384 R r r = = * 0 0.05 r = 0 0 R = 0 0 R ® t ® ¥ 3 0
t h t agreement with the general prediction from the diagrammatic expansion of quench chemical potential pointed out in section 5.3A.

Concluding remarks

In the present section, we studied in details the quench effect on the chemical potential of a hard-sphere fluid confined in a hard-sphere matrix. A quantitative measure is introduced which is the difference between the chemical potential of the confined fluid and that of the corresponding equilibrium binary mixture. From the diagrammatic expansion of quench chemical potential, we see that there is no quench effect at the linear order of matrix density. In some cases, e.g., one point fluid particle in an HS matrix or one finite size HS fluid particle in a matrix of point particles, both the chemical potential of the quench-annealed system and that of the corresponding equilibrium counterpart can be calculated exactly and analytically. In these cases of one single fluid particle, the quench effect vanishes strictly as well. Hence, the quench effect shows up only when fluid-fluid and matrix-matrix correlations are both present. Under more general conditions, we applied scaled particle theory for calculating the chemical potential of the quench-annealed system and that of its equilibrium counterpart.

Thorough numerical investigations and analytical analysis allowed us to identify the parameter which controls the appearance of quench effect for the system considered here. When matrix particles are larger than fluid particles, quench effect is small for physically meaningful matrix and fluid densities and the equilibrium binary mixture appears slightly more crowded than the quench-annealed system. SPT predicts that in the limit of extremely large matrix particles, i.e., , the quench effect vanishes. In the case of matrix particles smaller than fluid ones, the quench effect can become strong as matrix and fluid densities increase. In this case, the quench-annealed system can appear more crowded than the corresponding equilibrium binary mixture even when the density of each species is strictly the same in both systems respectively.

The comparison between SPT and MC results show that the fluid chemical potential of

1 0 / 0 R R t = ®
the quench-annealed system can be significantly overestimated by SPT in the case of fluid particles larger than the matrix particles. This is in sharp contrast with the excellent accuracy of SPT for confined HS fluid when the fluid particle is not larger than the matrix particle. For the moment, the origin of this inaccuracy does not appear clear and deserves certainly further investigations. Nevertheless, the trend for the variation of quench effect predicted by SPT is in agreement with that found from MC simulations.

Although the conclusions given above are obtained from the study of a simple model with only repulsions between hard cores, they should be general since the size effect exists for any real fluid and porous materials. Moreover, our results provide also some useful insights for the design of functionalized porous materials. For highly divided porous materials, i.e., those with its tiny solid constituents evenly distributed in space (e.g., aerogels), the quench effect is large and it can be hard to adsorb large fluid particles into such materials even when the porosity is high. For example, such a thumb rule can be useful for elaborating porous membrane for separation. Moreover, the findings of the present work should be useful also for assessing the quench effect on diffusion since the thermodynamics factor in an experimentally measured activitybased diffusion coefficient 266 is directly related to chemical potential 267 . Nevertheless, a detailed discussion of this interesting topic is far beyond the scope of the present work.

Final remarks

Porous materials have important applications in both fundamental research and chemical industry, such as oil exploitation, catalysis, capturing Greenhouse gas, energy storage devices, and design of sensors. For instance, Aerogels with their highly macroscopic operability, extremely surface area, and recoverability, are widely used in photocatalysis. Many of those applications are based on the behavior of fluid adsorption in porous materials, such as the drug release, and gas separation. Thus a thorough understanding of the behavior of fluid confined in porous materials is necessary.

During the last few decades, a large number of theoretical and experimental investigations of fluids confined in different kinds of porous material had been made.

The properties of confined fluid can be affected by the geometry of porous material, i.e., porosity, pore-connectivity, pore size distribution, pore shape, and disorder effect.

Unfortunately, we still do not know the respective roles played by those geometric properties. Different from the bulk fluid, while the fluid particles adsorbed in porous material, the fluid particles feel not only the interaction from the surrounding fluid particles but also that from solid ones. This additional solid-fluid interaction can vary from very repulsive to very attractive ones. The different solid-fluid interaction can make the confined fluid exhibit very different behavior even materials with same geometry. Although confined fluid appears more complicated due to the presence of complex confining environment of adsorbent, one can wonder if there is any connection between confined and bulk fluids. To answer this question, in Chapter 2, we derived a general scaling relation between the confined fluid and the corresponding bulk onecomponent fluid, which allows for connecting some thermodynamic properties of a confined fluid, i.e. chemical potential, Helmholtz free energy per particle, and grand potential per particle, to the bulk ones. The validity of this scaling relation is established with the help of a large number of simulation results, in a wide range of confining environments, from single isolated pore to random porous medium, and the fluid-fluid and fluid-matrix pair interaction, from pure repulsive to repulsive plus attractive interaction. Since this scaling relation holds for chemical potential, Helmholtz free energy per particle, and grand potential per particle, one can immediately find that if Helmholtz free energy per particle and grand potential per particle are expressed as a function of chemical potential, the confined fluid will show the same behavior with the corresponding bulk one. This invariance implies the chemical potential should be considered as independent variable instead of fluid density, in the study of the confined or inhomogeneous fluids. Although a derivation of the scaling relation from the firstprinciples is currently unavailable, we have revealed its intimate connection with general theoretical frameworks like Gibbs theory for inhomogeneous fluids and morphological thermodynamics. We believe this is why the scaling relation works so well under wide conditions and for a large variety of confining environments. The scaling relation shows clearly that the porosity (space accessibility) and fluid-solid interaction (through the adsorption term) is of primary importance for determining the thermodynamics of confined fluids. The other characteristics, like pore-connectivity, pore-shape, pore-size distribution, etc., play less significant role. One potential application of our scaling relation is circumventing some experimental difficulty for direct measurement of some thermodynamic properties of confined fluid. In this work, the scaling relation was only used to study some thermodynamics properties of confined fluid. One nature extension is to see the scaling relation still holds for dynamics properties, such as diffusion coefficient. Another intuitive extension is to consider if the scaling relation holds also for the fluid adsorption in flexible porous materials, e.g. metal-organic frameworks (MOFs) or to find the modifications needed if necessary.

For fluid adsorption in a real porous material, the fluid-solid interfaces are generally curved ones. It might appear surprising that the thermodynamics for dealing with curved interfaces is not so well established although early investigations go back to Tolman. Thus a thorough understanding of the interfacial properties is necessary.

Scaled particle theory is a powerful tool for studying the properties of fluid near to a curved wall. The original motivation for scaled particle theory is to derive a simple equation of state for a bulk hard sphere fluid. Then, it was recognized that scaled particle theory provides not only the thermodynamic properties of a hard sphere fluid in bulk but also the surface tension of a hard sphere fluid near a spherical hard wall. However, the accuracy of its predictions is not as high as other recent equation of state like Carnahan-Starling equation of state. Mecke and co-workers have made efforts to develop a general framework, named as morphological thermodynamics, to account for more complex surface morphology. According to Hadwiger theorem in the integral geometry, the morphological thermodynamics assumes that the thermodynamic potential of an inhomogeneous system can be expressed as a linear combination, which only contains four terms, i.e., one bulk contribution proportional to system's volume and three surface contributions proportional respectively to surface area, mean and Gaussian curvatures of the interface. It is to be pointed out that in scaled particle theory, the chemical potential for creating a spherical cavity inside a fluid is assumed to have this form. The foundation of this method has been questioned recently. The contribution from the terms beyond the four Hadwiger's terms has been revealed from both diagrammatic expansion approach and molecular simulation. It is natural to ask that is it possible to include non-Hadwiger terms in scaled particle theory? To answer this question, in Chapter 3, two augmented scaled particle theory were proposed. The first one contains an adjustable parameter, and the second one uses two matching processes to circumvent the adjustable parameter. Both of them have simple analytical expressions. Also, both of them do not only significantly improve the accuracy of thermodynamic properties, i.e., pressure and chemical potential, but also improve the accuracy of surface properties, i.e., planar surface tension and Tolman length. Also, the same idea can be used to treat the multi-component hard sphere fluid system, and the improvement is significant. The success of our two augmented scaled particle theory indicates that the morphological thermodynamic is not an exact theory, it is only a very good approximation. One intuitive extension of this work is to consider if the more non-Hadwiger terms are considered, will the higher accuracy of scaled particle be obtained?

For description of porous medium, many theoretical models for porous medium were carried out. Models with simple pore geometry, such as slit, cylinder pore, are widely studied. In such models, the pore-size distribution and pore connectivity are neglected. For ordered porous materials like MOF and ZIF, have been studied by molecular simulation and density functional theory. To consider the quench disorder effect, some random porous medium models have been carried out, i.e., Madden-Glandt matrix model, Van Tassel's templated matrix model, and hard sponge matrix model.

The equation of state for confined fluid strongly depends on the model of porous medium. So, is it possible to derive a general equation of state which is irrelevant with the model of porous medium? In Chapter 4, we derived a new general equation of state for describing the thermodynamic properties of fluids confined in different porous medium. To our best knowledge, this is the first equation of state for confined fluid that is irrelevant to the model of porous medium. In this general equation of state, the chemical potential is considered as the independent variable. In a large range, our new equation of state works well. Our new equation of state indicates that the porosity and other three geometric properties play the primary roles for the thermodynamic of confined fluid. All the other parameters, i.e., pore shape, pore connectivity, pore size distribution, and disorder effect, are much less important, which is similar to the results from our scaling relation. We also reported that even for an ideal gas fluid, in some cases, the morphological thermodynamic still does not hold, which means the morphological thermodynamics is only a good approximation. In this work, we only consider the hard sphere fluid. One nature extension of this work is to see if this equation of state still holds for other fluid, such as square-well fluid and Lennard-Jones fluid. An interesting application of this work is to consider if this equation of state can be used to describe the flexible porous materials.

The Madden-Glandt model is also called as a quench-annealed mixture with quenched species referring to the matrix and annealed species referring to the confined fluid. Although many investigations have been devoted to study fluids confined in the random porous medium since Madden-Glandt model was proposed, no precise measure has ever been proposed to describe quantitatively the quench effect for the adsorption of a fluid in a random porous material. In Chapter 5, we introduced a quantitative measure of quench effect which is the difference between the chemical potential of the confined fluid and that of the corresponding equilibrium binary mixture. With the help of diagrammatic expansion, we revealed that this is no quench effect at the linear order of matrix density. Hence, the quench effect shows up only when fluid-fluid and matrixmatrix correlations are both present. With the help of scaled particle theory for confined fluid and the molecular simulation, we found that when matrix particles are larger than fluid particles, quench effect can be ignored. When the fluid particles are larger than matrix particles, the quench-annealed system can appear more crowded than the corresponding equilibrium binary mixture even when the density of each species is strictly the same in both systems. This indicates that the ratio of the radius of fluid particles to that of matrix ones plays the primary role for the quench effect. The comparison between scaled particle theory and molecular simulation results show that the fluid chemical potential of the quench-annealed system can be significantly overestimated by scaled particle theory in the case of fluid particles larger than the matrix particles. This is in sharp contrast with the excellent accuracy of scaled particle theory for confined hard sphere fluid when the fluid particle is not larger than the matrix particle. In my opinion, this is caused by the definition of porosity of porous materials.

There are two porosities, geometric porosity, and probe particle porosity, in the scaled particle. By analyzing the contribution of each term, one can find that this overestimate comes from the probe particle porosity term. It means the probe particle porosity underestimated the fluid particle accessible volume of porous medium. If only the geometric porosity is used in scaled particle theory, SPT2a, the fluid chemical potential of the quench-annealed system will be underestimated. This can be explained by a simple example, consider fluids confined in a cubic lattice matrix which the distance between two matrix particles is smaller than the diameter of fluid particle, in this system the fluid chemical potential is approach to infinity, but the prediction of chemical potential from SPT2a is same as the corresponding the bulk ones. This means the geometric porosity overestimated the fluid accessible volume of porous medium. Hence, how to find an appropriate porosity is the key to improve the accuracy of scaled particle theory for confined system. Our results provide some useful insights for the design of functionalized porous materials. For highly divided porous materials, i.e., those with its tiny solid constituents evenly distributed in space (e.g., aerogels), the quench effect is large and it can be hard to adsorb large fluid particles into such materials even when the porosity is high. For example, such a thumb rule can be useful for elaborating porous membrane for separation.

As a final remark, we would emphasize that with the help of theoretical approach and molecular simulation, this thesis studies the thermodynamics properties of confined fluid, clarifies the control variables of confined fluids, and discovers the common law of thermodynamics properties among different confined fluids. A scaling relation and two new equations of state were reported, they deepened the understanding of confined fluid, and advanced the development of the thermodynamics for confined fluid.

( ) 1 4 1 0 2 3 s s s ex R s s s s s s s R R R R R e R P R R R R d h bµ p b f f f f - - ì é ù - - + -< £ ë û ï ï = í - ï + + + + < ï î ( ) ( ) ( ) 0 ln 1 0 ex ex s s s s R R bµ h bµ - + = =- -= = 0 0 1 3 1 s s ex ex s s s s R R R R R bµ bµ h h - + = = ¶ ¶ = = ¶ - ¶ ( ) ( ) 2 2 2 2 2 2 0 0 3 2 1 1 s s ex ex s s s s R R R R R h h bµ bµ h - + = = + ¶ ¶ = = ¶ ¶ - ( ) ( ) ( ) ( ) 2 3 3 3 3 3 3 3 3 0 0 6 1 7 9 1 1 1 1 1 s s ex ex s s s s R R P R R R R h h h h b r bµ bµ h h - + = = + + - ¶ ¶ = + = ¶ ¶ - - i f ( ) 1 0 ln 1 R R f f h d - = - -- 2 1 1 1 3 1 2 R R R f h f h d - é ù ae ö = + ê ú ç ÷ - è ø ê ú ë û 3 1 2 2 1 3 3 2 1 1 3 R R R f h h f h h d - é ù ae ö ae ö = + - ê ú ç ÷ ç ÷ - - è ø è ø ê ú ë û ( ) ( ) ( ) 2 3 4 1 5 2 5 24 9 9 2 1 2 1 1 1 P R R h h f d b h h h h r h h h - é ù - ae ö ae ö ae ö = - - - ê ú ç ÷ ç ÷ ç ÷ - - - - è ø ê ú è ø è ø ë û 0
( ) ( ) ( ) 2 3 2 3 1 6 9 ln ln 1 1 2 1 1 1 1 2 6 s R P R R R R R e R d h h b bµ b r h h h h r f d d d - - ae ö = F = = L - -+ + + ç ÷ - - è ø é ù ae ö ae ö + - -+ - ê ú ç ÷ ç ÷ è ø è ø ê ú ë û ( ) 3 11 3 2 1 1 ASPT SPT a h h bµ bµ d h h ae ö ae ö = - + ç ÷ ç ÷ - - è ø è ø ( ) ( ) 2 3 6 3 1 6 6 1 R a e R R R R d d d d d d é ù ae ö ae ö = - + - - ê ú ç ÷ ç ÷ è ø è ø ê ú ë û ( ) ( ) 3 2 11 1 1 1 6 1 2 1 1 3 1 3 1 1 1 ln 1 ASPT SPT P P a b b h h h h d r r h h h h h h ì é ù ae ö ae ö ï = - + + - ê ú í ç ÷ ç ÷ - - - - ê ú è ø è ø ï ë û î ü -- -ý þ ( ) ( ) ( ) 2
R R R R R R R R e R d g p b g g g g g - - - - ae ö = = + + ç ÷ è ø ae ö + - ç ÷ è ø ! ! ! ! ! 3 11 2 11 3 2 0 0 2 6 ASPT ASPT SPT R R g f g g d - ae ö = = - ç ÷ è ø ! ! ! 2 11 11 3 1 1 1 2 ASPT ASPT SPT R R g g f g d - - - ae ö = = + ç ÷ è ø ! ! ! 11 11 2 0 2 3 ASPT SPT ASPT R g f g g d - - - = = - ! ! ! (A.18)
By using the same way as presented in Chapter 3, the adjustable parameter is obtained as . Then we presented the numerical results of APST11 below. Following Figures show that ASPT11 significantly improved the accuracy of both bulk properties and interfacial properties. However, for non-Hadwiger term, the prediction of ASPT11 underestimated it. From the Hadwiger's theorem, should be equal to for . It means that if is expanded in powers of , and higher orders terms will not exist.

R R R R R e R P R R R R d h bµ p b f f f f - - ì é ù - - + -< £ ë û ï ï = í - ï + + + + < ï î i f ( ) 0 ln 1 f h = - - 2 1 1 1 3 1 R R R f h f h d - é ù ae ö = - ê ú ç ÷ - è ø ê ú ë û 3 1 2 2 2 1 3 3 2 1 1 R R R
R R R R R R R R e R
The expansion of as following, , (B.20)

It is found that the and higher order terms still exist. This proves that morphological thermodynamics is not an exact method and can only serve as an approximation. It should be pointed out that the interface between fluid and the solute particle is a concave interface from fluid view in Hansen-Goos' work, and from

Hadwiger's theorem, the interface must be a convex one. In the follow section, we will study a system in which the interface is a convex one from fluid view. Clearly, the non-Hadwiger terms ( and higher orders) still exist in this convex system. This proves that morphological thermodynamics is not an exact method and can only serve as a good approximation. This result shows that we still don't understand

X c R R R R R R R Y R R R R Q R p p p ae ö ¢ = - + + - + ç ÷ ¢ ¢ ¢ è ø ¢ ¢ ¢ ae ö ¢ - - + - + ç ÷ ¢ è ø ¢ ¢ ae ö + - + + ç ÷ ¢ è ø arctan( ) X Q = 3arctan Q y R p ae ö = - ç ÷ ¢ è ø 2 3 4 Q R¢ = - 1 h 3
Hadwiger's volume theorem, we need come back to the theorem and find out where is the morphological thermodynamics violate the restriction of this theorem.

( 1 )

 1 Nous avons étudié les propriétés thermodynamiques (potentiel chimique et la pression) d'un nombre important de fluides confinés dans divers milieux poreux. Une relation d'échelle générale a été trouvée, qui relie le potentiel chimique ainsi que la pression, des fluides confinés au celui du fluide en phase homogène. Cette relation d'échelle montre que la différence de propriétés thermodynamiques entre un fluide confiné et un fluide homogène peut être décrite uniquement par la porosité, la quantité d'adsorption en excès. Cette relation d'échelle a avancé significativement notre compréhension de la thermodynamique des fluides confinés et ouvre des perspectives très intéressantes pour déterminer les propriétés thermodynamiques de fluides confinés à partir des celles des fluides homogènes.

  : fluides confinés; matériaux poreux; thermodynamique statistique; 以构建适用于不同受限流体的普适状态方程为目标,通过结合形态热 力学与 SPT,探究了影响受限流体热力学的变量。通过将多孔材料的孔隙率,表

  of the well, and is the width of the well. Since the attractive part is from the spontaneous fluctuation, which means it should have a very smooth form. The Lennard-Jones potential offers a continuous function to describe the pair potential, ,

  In Kierlik's work, the Madden-Glandt matrix was obtained from Grand canonical ensemble. It should be mentioned that only mechanical pressure can be used in molecular simulation. In 2005, Dong and Chen first proposed a Virial equation for the thermodynamic pressure which can be used in molecular simulation and an exact relation between the pressure of fluids confined in porous material and the pressure of the corresponding bulk fluid. 164 This relation provides, for the first time, the basis of an experimental method for measuring

  Patsahan et al. first extended SPT to fluids confined in Madden-Glandt matrix model and obtained an accurate analytical equation of state. The basic idea of SPT is to insert an additional scaled particle into a fluid, this is equivalent to creating a spherical cavity. The excess chemical potential of inserting a point scaled particle can be obtained exactly. And a thermodynamical consideration can be used to describe the work of creating a finite size spherical cavity. Patsahan et al. introduced the exact analytical results of a point scaled particle in hard sphere or overlapping hard sphere matrix, and the first equation of state for confined fluids, SPT1, was carried out. However, in SPT1,

V

  condition in three space directions when a fluid confined in a matrix is considered. For slit pores, the simulation box is made with two square walls separated by a distance equal to and the periodic boundary condition is applied only in the two space directions parallel to the walls. For each simulation, about 2Í10 5 -1Í10 6 trial moves for each fluid particle are performed. Since finite-size matrices are used, any observable quantity fluctuates with matrix realizations and an average made typically with about 10 matrix realizations leads to converged results. The excess chemical potential of the fluid,

  .16), named as confinement-adsorption scaling, for a large variety of confined fluids under broad conditions. To show the general and robust character of this scaling relation, we considered virtually all types of confined fluids, i.e., in various porous environments like isolated slit pores, connected ordered or random porous matrices including sponge-like ones and different types of fluid-fluid and fluid-solid interactions. Fig. 2.1(a) shows the collapse of adsorption isotherms of an HS fluid confined in 19 different environments to the bulk isotherm after scaling according to eq. (2.18) while in Fig. 2.1(b), the results before scaling are presented. The robustness of the scaling relation is demonstrated by the large diversity of the considered confining environments, from isolated slit pores to random porous matrices with pure repulsive or repulsive plus attractive fluid-solid interactions. To establish the general validity of the scaling relation, we carried out also simulations with a Lennard-Jones (LJ) fluid (see results in Fig. 2.2). These results show the general character of the scaling relation which holds also for fluids with attractive interaction.

Figure 2 . 1 .Figure 2 . 2 . 2 . 4 .

 212224 Figure 2.1. Monte-Carlo simulation results for an HS fluid confined in 19 different confining environments: a) Chemical potential, , as a function of scaled density

Figure 2 . 3 .

 23 Figure 2.3. Monte-Carlo simulation results for an HS fluid confined in 19 different confining environments: a) Free energy per particle, , as function of scaled density

Figure 2 . 4 .

 24 Figure 2.4. Monte-Carlo simulation results for an HS fluid confined in 19 different confining environments: a) Free-energy per particle, , as function of chemical potential, ; b) Minus grand-potential per particle, , as function of chemical

  using theGibbs-Duhem equation. In 1975, Mandell and Reiss 249 made N = 5 and used five exact conditions to obtain an improved equation of state. However, the accuracy of this version is still limited, particularly at high densities. Recently, Corti and his co-workers 256 made N = 7 and used five exact conditions and two approximate conditions to modified SPT. However, since an integral equation contained in those conditions, this improved SPT does not have an analytical expression. They found that increasing the number of conditions does not lead to improvement of the predictive accuracy of the fluid properties by itself.

( 3 . 2 )

 32 , we account for this indistinguishability by ,(3.3) In eq.(3.2), is the insertion probability and the following exact expression holds,

  key importance of eq.(3.4) is that the higher order terms appear only successively when the radius of the scaled particle increases. For example, we have , fraction of hard spheres ( : density) and is the volume of the overlapping region of two spheres having a radius of and separated by a distance of , which is given by . (3.11) Although eqs.(3.2) and (3.3) indicate a possible road for obtaining chemical potential, one encounters quickly several major obstacles on this road. First, eq. (3.4) is an 0 1

  eq.(3.38) leads to an equation involving chemical potential, , and pressure, P. With the help of Gibbs-Duhem equation, one can eliminate either or and obtain a first order differential equation for P or . Because this is a differential equation with non-constant coefficients, it is not possible to solve it analytically. Nevertheless, this difficulty can be circumvented by an iterative procedure to obtain successively analytic results for chemical potential and pressure. One can start ( ) ( )

  result for chemical potential given in eq.(3.40) into Gibbs-Duhem equation and integrating the latter, we obtain the following ASPT1 result for pressure, ,

  radial distribution function and the Virial equation for the pressure of an HS fluid was used when going to the last equality in eq.(3.72). It is to note that eq.(3.72) yields immediately an expression for the pressure, i.e.,(3.74) At this stage, the above equation of state can be considered as the result of a new variant of scaled particle theory. As we will show in Section 3 that eq.(3.74) underestimates the pressure and gives less accurate results that the original SPT. It is emphasized that we will use eq.(3.64) and the coefficients given in eqs.(3.69)-(3.71) and(3.74) as the results of a first extrapolation of the chemical potential to insert a larger scaled particle with a radius in the region of .

  respect to density on both sides of eq.(3.84) and usingGibbs- Duhem equation, i.e., eq.(3.18), allow for obtaining a first order differential equation either for the chemical potential or for the pressure. Again, the resulting differential equation does not have constant coefficients and this prevents us from solving it analytically. By adopting the same iterative procedure as for ASPT1 [i.e., using the SPT value for pressure on the RHS of eq.(3.84), we obtain the following results (named as ASPT2) for chemical potential and pressure after the first iteration, ,

  of i-component fluid particle. One should be noted that should present, in principle, the radius of the smallest fluid particle. However, this restriction is not used in this paper. Eqs. (3.65)-(3.68) and (3

  .41) is as close as that of Carnahan-Starling equation, which has the only at high fluid densities, requiring at a high fluid density (with being chosen here) should allow for obtaining accurate values of pressure from ASPT1. This leads to the following equation and (3.41), we see that any positive root of eq.(3.100) smaller than can decrease the pressure given by SPT so that becomes possibly close to . Solving eq.(3.100), we find only one positive root in the considered region,. Alternatively, if we require at , we obtain the

  pressure (see Fig.3.1) and in the following discussions, we will use for ASPT1.

Figure 3 . 1 .

 31 Figure 3.1. Comparison of pressure of a hard sphere fluid. The CS results are presented by black squares, the red line is ASPT1 with , and the blue line is ASPT1 with .

Fig. 3 . 2

 32 Fig. 3.2 shows the comparison of the pressure, , obtained by ASPT1 and

Figure 3 . 2 .

 32 Figure 3.2. Comparison of pressure of a hard sphere fluid. The simulation results are presented by black squares, the red circle is SPT 7M , the green line CS, the blue line SPT 3 , the black line ASPT1, the red line ASPT2.

  prediction of pressure and surface tension becomes worse. Carnahan-Starling equation of state still has the best accuracy among those equations of state. The prediction from ASPT and ASPT2 are better than original SPT, especially ASPT2 almost has the same accuracy with Carnahan-Starling equation of state. One should note that all SPT results always overestimate or underestimate the Virial coefficients, but the prediction of Carnahan-Starling equation of state underestimates the fourth and fifth Virial coefficients and overestimates the others.

Fig. 3 . 2

 32 plots the planar surface tension of hard sphere fluid obtained from CS-SPT, CS-SPTM, original SPT, MFMT, molecular simulation, and two new ASPT. From Fig. 3.2, it shows all versions of SPT have the same trend with molecular simulation but the CS-SPT. CS-SPT appears a different trend with molecular simulation results in the high packing fraction. The original SPT overestimates the surface tension. Except CS-SPT and original SPT, all equations of state are nearly indistinguishable from the simulation results on the scale of the plot.

Fig. 3 . 2 .

 32 Fig. 3.2. Comparison of surface tension of a hard sphere fluid. The simulation results are presented by black squares, the green line MFMT, the blue line SPT, the black line SPT5, the red line ASPT2, the yellow line CS-SPT, the magenta line CS-SPTM

Figure 3 . 3 .

 33 Figure 3.3. Comparison of of a hard sphere fluid. The simulation results are

Figure 3 . 4 .

 34 Figure 3.4. Comparison of of a hard sphere fluid. The simulation results are

Fig. 3 . 5

 35 shows the prediction of non-Hadwiger coefficient from CS-SPT, CS-SPTM, ASPT1, ASPT2, SFMT, and molecular simulation. As it shows, the prediction from SFMT has the best accuracy. Except for SFMT, only ASPT1's prediction has the same magnitude with simulation result. ASPT2 significantly underestimates the Non-Hadwiger coefficient. Although the prediction from ASPT1 and ASPT2 have the same shape with simulation, the value of non-Hadwiger coefficient is larger than simulation results at least two orders of magnitude.

Figure 3 . 5 .

 35 Figure 3.5. Comparison of non-Hadwiger coefficient of a hard sphere fluid. The simulation results are presented by black squares, the blue line SFMT, the red line ASPT2, the black line ASPT1, the yellow line CS-SPT, the magenta line CS-SPTM

Figure 3 . 6 .

 36 Figure 3.6. Comparison of the excess chemical potential of a binary mixture system.

  find the expression of grand potential for an ideal gas fluid around a spherical matrix particle satisfies morphological thermodynamics expression, and the four coefficients are obtained. In different situations, such as ideal gas fluid confined in a

  obtained a general relation between confined fluid density and equilibrated bulk fluid density. Combine eq. (4.20) to eq. (4.24), the relation between confined fluid density, , and chemical potential, , can be obtained. This derivation is irrelevant with the model of porous medium, so we can use this model to study different confined fluid. It should be mentioned that scaled particle theory can also offer an accurate equation of state for different kinds of confined fluid like showed

Fig. 4 . 3 Figure 4 . 3 .

 4343 Fig. 4.3 (a)-(c) plots the prediction of chemical potential for an HS matrix from our new

Fig. 4 . 4 Figure 4 . 4 .

 4444 Fig. 4.4 shows the prediction of chemical potential for a hard sponge matrix in different porosity. As in the previous section, both our new equation of state and SPT2b1 have a great agreement with simulation results. In should be noted that when porosity is small, our new equation of state slightly underestimated the chemical potential in the high fluid density region. Compare Fig. 4.3 and Fig. 4.4, one can find that the pore shape is not important.

Fig. 4 .

 4 Fig. 4.6 shows the comparison of the chemical potential, , obtained by our new

4

 4

  the molecular simulation results on the scale of the plot in different pore width. Fig. 6 b) and c) shows the results of a hard sphere confined in a square well pore with different width of potential and different depth of potential. Those results indicate our new equation of state works well for fluid confined in slit pore system with different parameters. Compare these results and the results for previous models, one should find that the pore size distribution and pore connectivity are not very important.

  topologically distinct, simple, connected graphs which are composed of one white fluid point, some or no black fluid points, weighted by , or no bonds between pairs of fluid points, some or no bonds between pairs of matrix points, some or no of fluid particles and the various Mayer functions are defined as, ,

  topologically distinct, simple, connected graphs which are composed of one white fluid point, some or no black fluid points, weighted by , black

  thermodynamic properties of various bulk HS systems and has been extended recently to quench-annealed systems as well. In this Chapter, we will apply SPT to determine the chemical potential of an HS fluid in an HS matrix and that of the corresponding binary HS mixture in order to assess quench effect. Previous investigations have shown that the variant called SPT2b1 gives very accurate results

Fig. 5 . 3 .

 53 For the two matrix densities considered, the chemical potential of the confined fluid and that of the corresponding equilibrium mixture are nearly the same even at high fluid densities. The mathematical reason for the moderate values of the quench chemical potential in the region of becomes clear by examining eq.(5.20). We see readily that no value of in this region can induce large variations of the quench chemical potential for physically meaningful values of matrix and fluid densities (

Figure 5 . 3 .

 53 Figure 5.3. Comparison of the excess chemical potential as a function of fluid density

  one observes , We will see in the next subsection that the quench effect becomes stronger in the region of . So, is the turning point for the quench effect becoming significant and also the point at which the quench chemical potential changes sign, i.e., from to . In Fig.5.4, the MC simulation results are also presented. For the equilibrium binary mixture, SPT gives always results in good agreement with the MC ones while it overestimates the fluid chemical potential of the quench-annealed system when the quench effect becomes significant.

Figure 5 . 4 .

 54 Figure 5.4. Comparison of the excess chemical potential as a function of fluid density

Figure 5 . 5 .

 55 Figure 5.5. Comparison of the excess chemical potential as a function of fluid density

  These results show that the quench effect becomes stronger when the size ratio of the fluid particle to the matrix particle increases.

Figure 5 . 6 .

 56 Figure 5.6. Comparison of the excess chemical potential as a function of fluid density

Figure A. 1 .Figure A. 2 .Figure A. 4 . 2

 1242 Figure A.1. Comparison of pressure of a hard sphere fluid. The simulation results are presented by the blue line, the black line is original SPT, and the red line is ASPT11

  noted that the sign of is different from the ones when n is an odd number and is same as the original SPT ones. From eqs.(A.20)-(A.23), we obtain the following expression for the chemical potential, . (A.24)By using the same iterative procedure as presented in Chapter 3, the results of chemical potential and pressure can be obtained, named as

  same way as presented in Chapter 3, the adjustable parameter is obtained as . Then we presented the numerical results of APST12 below. Since the results of in ASPT12 is same as the original ones, we only presented the numerical results of P and (i = 0, -1, -3). One interesting find is the prediction of the non-Hadwiger term has a different sign with the result with an odd n.This indicates a potential form for describing non-Hadwiger terms, i.e.,. With the suitable parameters, and , the prediction of non-Hadwiger term might have the same trend as the ones form molecular simulation.

Figure A. 6 . 2 dFigure A. 7 .Figure A. 8 .

 6278 Figure A.6. Comparison of pressure of a hard sphere fluid. The simulation results are presented by the blue line, the black line is original SPT, and the red line is ASPT12

  Figure A.9. Comparison of of a hard sphere fluid. The simulation results are

  {sum of all the connected diagrams which are composed of one white fluid point, some or no black fluid point, wighted by , one white cavity point connect with each fluid point, the connected diagrams which are composed of one white fluid point, some or no black fluid point, wighted by , some or no bonds between pairs of fluid

  PMF) 。最近形态热力学的基础被广泛质疑。Laird 223 使用分子动力学 模拟(Molecular Dynamics simulation, MD)计算了流体在球形和圆柱形表面的表面 张力。通过使用多项式拟合 MD 结果发现不仅仅存在式(C.5)中的四项,还有关于 Hadwiger 定理仅仅在凸表面时成立。那么自然 而然可以提出以下问题:是否可以通过将 non-Hadwiger 项加入到 SPT 理论中来 进一步提升 SPT 理论的精度?是否可以找到一个简单的方法来加入 non-Hadwiger 项,来使得 SPT 的结果仍然具有完全解析和具有简单表达式的性质? 在第三章中,提出了两种 SPT 的改进版本(Augmented Scaled Particle Theory, ASPT) 。在 ASPT1 中,包含了一个可调参数,, Holovko 等人提出了一个基于 SPT 理论的解析的状态方程来描述流体在 Madden-Glandt 模型中的热力学性质 66,195 。

  Glandt matrix. This process can simplify the description of the structure of templated porous materials. The Ornstein-Zernike equations of fluids confined in Zhao's template matrix model were proposed by Zhao et al. by using both diagram expansion and replica method. Normally, there are two methods for studying the properties and the behavior of fluid confined in porous medium, simulation, and theoretical approach.

The template components are removed by heat treatment, acid or alkali solution. The pore size of this kind of material can be from micropore to macropore, two models cannot describe. The first templated matrices model is reported by Van Tassel. The Van Tassel matrix model is generated by fixing a two-component equilibrium fluid one component is considered as matrix species and the other component is the template species. After fixing, the template component will be removed. The pores in this model are generated from two parts, the first one is the voids between matrix particles, the primitive one, and the second one is from the removed template species. This model is very similar to the process of fabricating the templated porous material in experiments. However, this process is hard to describe in theoretical study. Even for an ideal gas fluid in Van Tassel matrix model, the expressions of the thermodynamics properties are not entirely analytically.

173

Zhao et al. also reported a templated porous medium. 175 In Zhao's template matrix model, the template pore is generated by piercing some cavities in the

Madden-

Table 2 .

 2

1. Computation parameters for canonical or grand-canonical ensemble Monte-Carlo simulations

Table 3 . 1 .

 31 Virial coefficients for hard sphere fluid

	n	Exact	SPT	CS	ASPT1	ASPT2
	2	4	4	4	4	4
	3	10	10	10	10	10
	4	18.3647	19	18	18.6526	18.5595
	5	28.2245(10)	31	28	29.3325	29.1675
	6	39.81545(15)	46	40	41.368	41.5656
	7	53.3418(15)	64	54	54.0742	55.631
	8	68.534(88)	85	70	66.7614	71.3043
	9	85.805(58)	109	88	78.7375	88.5556
	10	105.8(4)	136	108	89.3093	107.369

Table 3 .

 3 any other equations of state result. ASPT1 and ASPT2 improve the prediction of fourth Virial coefficient compare with original SPT. Although CS-SPT does not have the same trend with molecular simulation result in high packing fraction, the first four virial coefficients are more accurate than original SPT, which means in the high packing fraction, the planar surface tension is dominant under the higher Virial coefficients.

	0 ! g	3 h = +	15 2	h	2	+	759 70	h	3	+O	( ) 4 h
	2 shows the comparison of the Virial coefficients obtained from MFMT,
	CS-SPT, CS-SPTM, original SPT, ASPT1, and ASPT2 with exact values. Table 3.2

shows that all equations of state can offer the exact first two Virial coefficients of planar surface. For fourth Virial coefficient, the result from CS-SPTM is slightly better than

Table 3 . 2 :

 32 Virial coefficients of planar surface tension

	n	Exact	CS-SPT	CS-SPTM	SPT	MFMT	ASPT1	ASPT2
	2	3	3	3	3	3	3	3
	3	7.5	7.5	7.5	7.5	7.5	7.5	7.5
	4	10.842	11.667	10.854	12	10.333	11.531	11.257
	When a hard sphere fluid in contact with a curved surface, the surface tension
	should be written as following,
						,		(3.103)
	where	is the radius of a spherical wall,		,	is the Tolman
	length and	is the bending rigidity. Since the prediction of	and	from
	ASPT1 are identical with original SPT, in the following discussion, we will only
	consider original SPT. Fig. 3.3 plots the prediction of	from MFMT, original SPT,
	CS-SPT, CS-SPTM, ASPT2, and molecular simulation results. Fig. 3.3 shows that all
	equations of state have a good agreement with simulation results but the CS-SPT.
	Compare CS-SPT and CS-SPTM, one can find that using two Laurent series to describe
	the reversible work will significantly improve the predictive accuracy of surface
	properties. Compare with original SPT, MFMT, and ASPT2 slightly improve the
	prediction of

. One should be noted that CS-SPTM has a different trend with the other equation of state. In moderate packing fraction, the prediction of CS-SPTM is

Table 3 .

 3

			g -!	1
							g -!	1
	g	1 -	3 1 h é	h	ae	8 27 3 35 16 p	ö	h	2	ù	( ) 4 h

3 shows that all equation of state can provide exact first three Virial coefficients of . The prediction of the fourth Virial coefficient from CS-SPT has a different sign

Table 3 .

 3

				3: Virial coefficients of
	n	Exact	CS-SPT	CS-SPTM	SPT	MFMT	ASPT2
	2	3	3		3	3	3	3
	3	3	3		3	3	3	3
	4	3.477	-3.333	0.903	3	3.333	3.172
	In the final part of this section,	are considered. Since the prediction of
	from CS-SPT and CS-SPT5M are terrible, we will not compare them here. Fig. 3.4 plots
	the prediction of	from MFMT, ASPT2, and molecular simulation. As shown in Fig.
	3.4, except for ASPT2, all equation of state has the same accuracy in	. The
	prediction from ASPT2 slightly improves the original SPT's result. As previous part,
	here, the Virial coefficients of	will be considered. Urrutia 224 also reported the first
	four Virial coefficients as following,
							.	(3.105)
	Numerical results of Virial coefficients are shown in Table 3.4. It shows that all
	equations of state can provide exact first three Virial coefficients, and all equations of
	state overestimate the fourth virial coefficients. Hence, the future study in	is
	necessary.

Table 3 . 4 :

 34 Virial coefficients of

	1

2

3 1 2 1 1 w R

I would also like to acknowledge to Campus France for an Eiffel scholarship, to Région Rhône-Alpes (France) for a CMIRA scholarship, and to the

C. Ordered porous medium

In the previous section, we considered our new equation of state for different kinds of disordered porous medium. In this section, the ordered porous medium will be considered. Here, we use a cubic lattice model to describe the ordered porous medium, the matrix particles placed in the lattice points. The four geometric properties are same with HS matrix. Fig. 4. 5 shows the comparison of the chemical potential, , obtained by our new equation of state with the results of molecular simulation. Our results are very close to the simulation values even in a high matrix density. The accuracy of our new equation of state will decrease while the increasing matrix density and the decreasing radius of the matrix particle. In high fluid density, our new equation of state is overestimating the chemical potential. Compare this result with the HS matrix one, we found when matrix particle is larger than the fluid particle, the disorder effect is not important.

Concluding remarks

In the present section, we introduced the morphological thermodynamic with the help of ideal gas fluid, we present even for an ideal gas fluid, in some cases, the morphological thermodynamic still does not hold. A new general equation of state is obtained from morphological thermodynamic.

D. Strong quench effect

Now, we consider the case that matrix particles are smaller than the fluid particles, i.e., . The results for and three matrix densities ( : 0.05, 0.2 and 0.3) are presented in Fig. 5.5. At very low matrix density ( ), the quench effect remains small (see Fig. 5.5a). As the matrix density increases, the quench effect becomes significant and increases with the fluid density (see Fig. 5.5b and 5c). At high fluid densities, is larger than (note that for , at high fluid densities). From Fig. 5.5b and 5c, we see that the fluid chemical potential of the quench-annealed system is overestimated by SPT compared the MC simulation results.

Nevertheless, the trend for the variation of the quench effect described by SPT is in agreement with that from MC simulations. It is to be pointed out that the accuracy of SPT for confined HS fluids has been checked previously only in the case of , i.e., the fluid particles being not smaller than the matrix particles. The present work confirms the good accuracy of SPT in this case but reveals that in the case of fluid particle smaller than the matrix particle, the accuracy of SPT deteriorates when fluid and matrix densities increase.

and .

(5.25)

We see now that and are different only by the third terms on the RHS of eqs. (5.24) and (5.25). This third term for increases exponentially with the matrix density while that for increases linearly. This explains why the difference between and increases very quickly with the matrix density. The exponentially-increasing term comes from the term in eq. (5.11).

Numerical results for finite values of in the region of show that the dominant contribution to quench chemical potential comes always from the term .

Moreover, it is worthwhile to note that eqs. (5.24) and (5.25) lead to a quite simple expression for the quench chemical potential, i.e., .

(5.26)

In the case of very large matrix particles, i.e., , the quench chemical potential varies linearly with with a slope depending only on the matrix density, , and the fluid particle size, . It is also to note that the RHS of eq. (5.26) does not contain the contribution of the linear term in matrix density and this is in perfect From morphological thermodynamic, the grand potential of this system can be expressed as eq. ( 4.11) with same parameters, which all same as expression above but the last term. The last term of expression above denotes the contribution of the region suffered interaction from both two solutes. This calculation proves morphological thermodynamic cannot use for when such overlapping part appears. The future study of morphological thermodynamic and Hadwiger's volume theorem is still needed.

Consider a binary mixture system, 1 and 0 are used to represent two components.

B. Non-Hadwiger coefficient in convex system

The system is a hard sphere fluid confined in a spherical hard sphere cavity, and the interaction potentials in this system are described as following, , where is the activity of fluid