Modeling Adsorption Properties on the Basis of Microscopic

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Modelling adsorption properties on the basis of microscopic, molecular and structural descriptors for non-polar adsorbents Edder J García, Javier Pérez Pellitero, Christian Jallut, and Gerhard D. Pirngruber Langmuir, Just Accepted Manuscript • DOI: 10.1021/la401178u • Publication Date (Web): 27 Jun 2013 Downloaded from http://pubs.acs.org on July 11, 2013

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Modelling adsorption properties on the basis of microscopic, molecular and structural descriptors for non-polar adsorbents

Edder J. García,‡ Javier Pérez-Pellitero,‡ Christian Jallut,║ Gerhard D. Pirngruber‡* ‡

║

IFP Energies nouvelles, Rond Point échangeur de Solaize, 69360 Solaize, France.

Université de Lyon, Université Lyon 1, Laboratoire d’Automatique et de Génie des

Procédés, UMR 5007, CNRS—ESCPE, 43, Bd du 11 Novembre 1918, 69622 Villeurbanne cedex, France. Keywords: Ruthven Statistical Model, curvature, statistical thermodynamics, descriptor, structure-property relationship

Abstract We propose a method for analytically predicting single component adsorption isotherms from molecular, microscopic and structural descriptors of the adsorbate - adsorbent system and concepts of statistical thermodynamics. Expressions of the Henry constant and the heat of adsorption at zero coverage are derived. These functions depend on the pore size, pore shape, chemical composition and density of the adsorbent material. They quantify the strength of the solid-fluid interaction, which governs the low pressure part of the adsorption isotherm. For 1 ACS Paragon Plus Environment

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intermediate and high pressures, the fluid-fluid interactions must also be taken into account. Both solid-fluid and fluid-fluid interactions are combined within the framework of the Ruthven Statistical Model (RSM). The RSM, thus, constructs theoretical adsorption isotherms, that are entirely based on microscopic molecular and structural descriptors. The theoretical results that we obtained are compared with experimental data for adsorption of pure CO2 and CH4 on all-silica zeolites. The developed methodology allows estimating the optimum properties of a non-polar adsorbent for the adsorption of CO2 in cyclic adsorption processes.

1. Introduction The rational design of porous solids as adsorbents for gas separation processes requires a detailed comprehension of the relationship between the microscopic properties of the solid and its macroscopic adsorption and separation properties. Such a rational approach becomes ever more important in view of the enormous progress recently made in the synthesis of zeolites, Metal Organic Frameworks (MOFs) and carbon materials. Since the number of available porous solids is exponentially increasing it becomes almost impossible to carry out an exhaustive experimental screening of all existing materials. Reliable methods for preselecting the most interesting materials are, therefore, urgently needed. Vice versa, material science has made the first steps towards a real design of porous solids, i.e. tuning the pore size and pore structure at will. If we were able to define the optimal pore structure for a given application, synthesis chemists could concentrate their efforts on the most promising materials. In both cases, i.e. for efficiently screening existing solids and for sketching the image of the ideal solid, we need to understand how macroscopic adsorption properties are linked to the microscopic properties of both the adsorbent and the adsorbate. Several approaches have been 2 ACS Paragon Plus Environment

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used in the literature to establish such a relationship. One of the most common methods is to perform molecular simulations of adsorption. Force field simulations require a complete microscopic description of the adsorbent (atomic positions, intermolecular potentials, etc.). Provided that the intermolecular potentials are correctly described, the force field simulations yield quite accurate results, because the porous solid is described in all its atomistic details. The main drawback of this atomistic description is the difficulty to identify general trends and patterns in the behavior of different adsorbents.

At best, empirical correlations can be

generated by carrying out molecular simulations on a large number of solids,1-4 which is, however, very time-consuming. Moreover, it is impossible (or at least very difficult) to invert the process, in order to sketch the picture of the ideal solid; the application of reverse Monte Carlo methods to adsorption is still mainly reserved to disordered porous carbons.5 In order to improve our understanding of adsorption properties of solid materials, we have to reduce the number of parameters describing the porous solid, at the sacrifice of some precision.6 Therefore, in the present work we derive a simplified and analytical adsorption model that explicitly makes the link between a few microscopic properties of the adsorbentadsorbate system and the adsorption isotherm. The key descriptors of the system are (1) the solid's pore size and shape through the concept of its pore curvature,7-12 (2) the strength of the adsorbent-adsorbate (solid-fluid) interaction, (3) the solid's density, (4) the strength of the adsorbate-adsorbate (fluid-fluid) interaction.

2. Construction an adsorption isotherm model from a few microscopic descriptors of adsorbate and adsorbent 2.1. General strategy In order to address the problem step by step, we divide an adsorption isotherm into three regions (Figure 1). The initial, low pressure part of the isotherm, i.e. the Henry region, is 3 ACS Paragon Plus Environment

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mainly governed by the strength of the solid-fluid interactions. If the interaction between an atom of the solid and of the adsorbate is described by a Lennard-Jones potential, the strength of the interaction is indicated by the depth of the potential well, εsf. The adsorbate atom does, however, not only interact with a single (the nearest) atom of the solid, but with all atoms of the solid that are within a certain cut-off distance (beyond the cut-off distance the interaction potential becomes negligible). The amplification of the solid-fluid interaction by the multiple interactions of the adsorbate with all the atoms of the solids depend on the atomic density of the solid. The adsorption potential is further amplified if the pore surface is curved around the adsorbate molecules. The higher the curvature, the stronger is the adsorption.13,14 This phenomenon is called the confinement effect in adsorption. Overall, the initial slope of the isotherm should mainly depend on three parameters: εsf, the solid's skeletal density and the pore curvature. When the adsorbate density increases, i.e. in the intermediate part of the isotherm, fluidfluid interactions become non-negligible. The magnitude of these interactions is related to intrinsic microscopic properties of the fluid such as its polarizability, permanent electric moments and the molecule´s shape. Fluid-fluid interactions can be quantified by choosing an appropriate equation of state (EoS). Simple equations of state, like the van der Waals equation rely only on two parameters: the fluid-fluid attraction εff and the excluded volume of the molecule, b. Finally, the quantity adsorbed at saturation of the isotherm is simply limited by the space available for the adsorbate molecules, i.e. by the relation between the pore volume and the adsorbate's excluded volume. Altogether, the entire adsorption isotherm is controlled by a fairly small set of microscopic parameters (Figure 1). By using statistical thermodynamics, it should be possible to construct

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an adsorption model based on these microscopic descriptors. This comes down to finding an expression for the partition function of the adsorbed phase Qads.15-20

qsat Adsorbed amount

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Saturation zone available pore volume adsorbate's excluded volume Transition zone additional fluid-fluid interactions Initial slope (Henry region) Solid-fluid interactions Skeletal density Pore diameter and shape (confinement)

Pressure Figure 1. Schematic representation of a type I or type V adsorption isotherm

Qads =

1 N ! Λ3 N

∫ exp(− NβE

sf

(r ) )exp(− NβE ff (r ) )dr

Eq. 1

v

N is the number of adsorbed molecules. β = 1/kT, v the volume available for adsorption and Λ the de Broglie wavelength, Esf and Eff are the solid-fluid and fluid-fluid interactions respectively.

2.2. Choice of the thermodynamic model In order to simplify the calculation of the partition function, solid-fluid interactions Esf are usually separated from fluid-fluid interactions Eff and the latter are treated by a mean field theory, i.e. they are calculated from an equation of state at the average density of the adsorbed phase.21 Qads =

1 exp( − Nβ E ff ) ∫ exp( − Nβ E sf (r ))dr N ! Λ3 N v

Eq. 2

Knowing the solid-fluid interaction potential, one can calculate the partition function of the adsorbed phase as a function of its density by resolving the above integral, then deduce the 5 ACS Paragon Plus Environment

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chemical potential of the adsorbed phase and finally calculate the equilibrium partial pressure in the gas phase. This leads to an implicit expression of the adsorption isotherm, i.e. the pressure as a function of the adsorbed phase density. In contrast to this approach via the partition function and the chemical potential, Ruthven has derived an explicit analytical expression of the adsorbed amount as a function of pressure,22,23 which is more convenient for our purposes. The Ruthven Statistical Model (RSM) is a so-called lattice model, i.e. it describes the adsorbent as a lattice of adsorption sites. The adsorption site in the RSM is an entire pore cavity, which can be filled with a certain number of adsorbate molecules (limited by the volume of the cavity). Ruthven's model assumes that the adsorbate molecules can move freely within a pore cavity. In other words, the adsorbed phase in the RSM is perfectly homogeneous, i.e. a distinction between a first zone that is in interaction with the solid and a second zone where fluid-fluid interactions dominate, as in the BET model24 or the model proposed by Travalloni et al.,20 is avoided. The strength of the solid-fluid interaction is described by a Henry constant. Fluid-fluid interactions are accounted for by a (mean-field) Sutherland potential, which leads to the van der Waals equation of state. The saturation capacity is given by the ratio between the volume of the cavity and the adsorbate's excluded volume. An important asset of the RSM is that it can be readily extended to multi-component adsorption, which is a necessary condition for treating separation phenomena. The RSM can depict a large number of experimental single and multi-component adsorption isotherms25-27 by using Henry constants and fluid-fluid intermolecular parameters fitted from experimental data. The novelty of our approach is to replace the fitted Henry constant by a value that is derived theoretically on the basis of the microscopic descriptors of the system. This is done by using idealized pore shapes such as cylinders or spheres to represent the cavities.6 In these idealized pore geometries the global solid-fluid potential can 6 ACS Paragon Plus Environment

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be treated analytically, provided that the interaction between adsorbate and adsorbent atoms can be described by a Lennard-Jones potential. We will show that this method allows reproducing experimental adsorption isotherms of non-polar materials, i.e. materials where dispersion-repulsion interactions govern the adsorption, without a single adjustable parameter. Having thereby validated the descriptor-RSM model, as we will call it from here on, we go on using it to sketch the properties of the optimal adsorbent, as a function of the operating conditions of the CO2 adsorption/separation process. 3. Modeling methods 3.1. The Ruthven Statistical Model The RSM describes the adsorbent as an ensemble of independent regular cavities having a maximum loading where the adsorbed phase is formed by freely mobile interacting molecules. This representation of the adsorption system is appropriate for describing the physisorption of small molecules in zeolites and MOFs, since even in the case of CO2 adsorption on polar zeolites an important degree of mobility of the adsorbed phase has been reported.28,29 As mentioned above, the contribution to the configurational integral is divided in two parts: (i) the solid-fluid interactions, characterized by the Henry constant KH and (ii) the fluid-fluid interactions that are treated by a mean field theory. The single component RSM equation is smax  ( K ⋅ p) s KH ⋅ p + ∑ H Z ff  1  s = 2 ( s − 1)! θ= smax smax  ( K H ⋅ p) s 1 + K ⋅ p + Z ∑ H  s! s =2 

Eq. 3

where p is the pressure, θ is the coverage (fraction of the cavity that is occupied) and smax is an integer defining the maximum number of molecules per cavity such that smax ≤ v /b. b is the excluded volume of the adsorbate and v is the volume of the cavity.


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The Henry constant is given by

K H = β ⋅ Z sf = β ⋅ ∫ exp[− E sf (r ) β ] v

Eq. 4

where β = 1/kT, Zsf is the solid-fluid configurational integral, and Esf is the solid-fluid intermolecular potential. The coefficient Zff accounts for the fluid-fluid interactions. It is given by Z ff

s  s 2 bε ff  sb  = 1 −  exp β ⋅  v  v  

   

Eq. 5

where εff is the depth of the fluid-fluid potential well. On the one hand, if the number of molecules per cavity is one the model reduces to the Langmuir equation. On the other hand, for a large number of molecules the RSM tends to the Volmer isotherm.23 The RSM equation is based on the idealization of the adsorption system as a collection of molecules adsorbed in isolated cavities. This discretization of the system leads to a saturation capacity that is an integer (smax). This is a drawback since most of the adsorbents contain a fractional number of molecules per cavities at saturation. The deviation is more important for small cavities. This problem can be solved by defining the adsorption site itself as an ensemble of several cavities, as Ruthven et al. suggested.22 Additionally, this modification allows taking into account the exchange of molecules between cavities. If the molecules are really highly mobile, this treatment improves the representation of the entropy of the system. It can be shown that the difference between the adsorbed amount in the single and multicavity form of the RSM is small when each cavity adsorbs more than five molecules (see supporting information). This condition is always fulfilled for small adsorbates like CH4 or CO2. As a consequence, the notion of the cavity as an adsorption site, although essential for the derivation of the RSM, finally becomes obsolete in its practical application. The maximum number of adsorbed molecules smax is not necessarily linked to the actual volume of the (multi-)cage, but becomes an arbitrary number that corresponds to the number of 8 ACS Paragon Plus Environment

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coefficients used in a virial expansion of the Grand canonical partition function.15 This is very convenient for our purpose, because it allows applying the RSM to channel-like pore systems where the notion of a cavity does not exist. In this work we have arbitrarily fixed smax to 120 and defined the volume of the arbitrary cavity as v = smax * b. We have further normalized the Henry constant by the volume of the cavity in order to avoid that its value artificially depends on smax. K=

KH KH = v smax ⋅ b

Eq. 6

Hence, Eq. 3 becomes s s max   s 2 ε ff   ( Kp ) s (bs max ) s  s    1 −  exp  Kp (bs max ) + ∑  s max kT   ( s − 1)! s=2 1   s max    θ=   s 2 s s max s max   s ε ff   ( Kp ) (bs max )  s   1 −  exp 1 + Kp (bs max ) + ∑  s max kT   ! s s s =2 max     

Eq. 7

3.2. The solid-fluid interaction potential 3.2.1. Geometrical representation of the adsorbent cavities In order to link the RSM to microscopic descriptors of the adsorbent the solid-fluid potential Esf and thereby also the Henry constant have to be expressed as a function of these descriptors. The solid-fluid potential is a three-dimensional function of the position of the guest molecules in the adsorption volume. In molecular force field simulations, the potential is numerically calculated by adding pairwise the Lennard-Jones potential of each atom of the solid with the adsorbate. For simple pore geometries, i.e. spheres and cylinders, it is possible to derive analytical expressions for the solid-fluid potential if the adsorbate is described as a single Lennard-Jones center.30,31,32 In reality, hardly any solid cavities correspond to perfect spheres or cylinders. We propose to represent the pore volume of a real solid by an assembly 9 ACS Paragon Plus Environment

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of spheres or cylinders We therefore have to find a method to transpose the real pore geometry to a simplified shape, i.e. an assembly of spheres or cylindres, having an equivalent pore radius R that exerts the same confinement effect as the real solid. In a separate work,33 we have developed a technique to calculate this equivalent pore diameter from the mean curvature of the accessible pore surface of the real solid. We will show here that if the equivalent spherical or cylindrical pore geometry is used within the framework of the RSM, experimental isotherms can be very well reproduced. In the following sections we derive adsorption potential and Henry constant within cylindrical and spherical pores. 3.2.2. Solid – fluid potential within a sphere To calculate the solid-fluid interaction in a spherical cavity all the contributions of the atoms in the surface must be added. This is done by integrating the intermolecular interactions over the surface of the sphere. This method leads to an analytical solution when the surface density of atoms ρsur is assumed to be constant and the Lennard-Jones (6-12) model is used to represent the dispersion-repulsion potential (Esf). The resulting equation gives the interaction of a single molecule with a layer of the solid as a function of the reduced radial distance x = r/R as follows:32

  1 2 * 1 1  1  1 1    E sf’ , sph x, R * = 8πR * ρ sur ε sf   (1 − x )10 − (1 + x )10  − 4 R *6 ⋅ x  (1 − x )4 − (1 + x )4 *12 10 R ⋅ x   

(

)

   

Eq. 8

If possible, the parameters in Eq. 8 were reduced with respect to the solid-fluid collision diameter σsf, i.e. we defined a reduced surface density as ρsur* = σsf2ρsur and a reduced pore diameter as R* = R/ σsf. The pore radius R is the distance between the center of the pore and the center of the atoms in the pore wall. r is the distance from the center of the pore. εsf is the depth of the potential well. 10 ACS Paragon Plus Environment

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If we only look at the attractive part of the Lennard-Jones potential in a spherical pore,

(

)

* E sf’ , sph ,att x, R * = −2π ⋅ ρ sur ε sf ⋅

1 R *4

1  1 1   −   4 4   x  (1 − x ) (1 + x ) 

Eq. 9

one can see that it is directly proportional to the product ρ*surεsf, which we call surface energy density (SED). We will later use a dimensionless surface energy density, that is defined as SED* = SED / kT . 3.2.3. Solid – fluid potential within a cylinder In the case of a cylindrical cavity, the expression for the solid-fluid potential is given by:30,31   9 9  3 3  2 F2 − ,− ;1; x 2    63 F1 − 2 ,− 2 ;1; x    −3  2 2  * E sf' ,cyl ( x, R * ) = πρ sur ε sf  10 10 4 4 *10 *4 R [1 − x ] [1 + x ]   32 R [1 − x ] [1 + x ]    

Eq. 10

where F1 and F2 are hypergeometric functions. In order to facilitate the calculations, these functions can be approximated by a small number of terms of Taylor-series expansion (see supporting information) in the range of 0 to 1 according to: F1 ≈ 1 +

81 2 3969 4 11025 6 ⋅x + ⋅x + ⋅ x + ... 4 64 264

Eq. 11

9 2 9 4 ⋅x + ⋅ x + ... 4 64

Eq. 12

F2 ≈ 1 +

3.2.4. Representation of the bulk solid In the potential equations 8 and 10, the pore wall is considered as a single layer of atoms, which form the pore surface. This approach ignores the contribution of the other atoms of the solid, which are not part of the pore surface, but located in more distant "layers". Hence, the solid-fluid potential will be underestimated. The opposite approach is to consider that the pore is surrounded by an infinite number of cylindrical or spherical layers, separated by a distance ∆, see Figure 2.34,35 This representation is also not correct; in reality the layers surrounding


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the pore are not dense, but interrupted by neighboring pores. In order to solve that problem we used the following heuristic: the critical parameter is the density of the solid atoms surrounding the pore. We, therefore, used the representation of Figure 2, but replaced the real distance between the solid layers by an "equivalent" distance, calculated from the third root of the solid's density. In other words, we redistributed the solid atoms in a homogenous way around the pore. In this way, the porosity of the solid is taken into account, i.e. the fact that dense pore walls alternate with neighboring pores. Eq. 13

∆ = 3 1 / ρ a = 3 Vuc / N s

ρa is the density of atoms in the solid, Ns is the number of atoms in the unit cell and Vuc is the volume of the unit cell. The overall solid-fluid potential was then expressed as the sum of the interactions of the guest molecules with each layer as follows: E sf ,tot ( x, R * ) =

mmax

∑E m=0

sf

( x, R * + m ⋅ ∆* )

Eq. 14

where m is the index of a given layer and ∆* = ∆/σsf. In practice, this sum converges rapidly with the number of layers.35 The parameter mmax plays the same role as the cutoff radius does in molecular simulations. In this work mmax was fixed at 10.

m=3 m=2 m=1 Δ

Figure 2. Illustration of a multilayer pore wall.

3.4. Calculation of the Henry constant from the solid-fluid potential


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The Henry constant is obtained from the configurational integral over the whole volume of the spherical or cylindrical cavity.35-38 To transform it into an intensive variable (which does not depend on the size of the system) it is then normalized by the accessible volume of the cavity vacc. In spherical symmetry, the normalized Henry constant becomes:39

K sph = β ⋅

1

4πR 3 exp(− β ⋅ E sf , sph ( x) )⋅ x 2 ⋅ dx ∫ vacc 0

Eq. 15

In the same way, for a cylindrical cavity the Henry constant is given by

K cyl = β ⋅

1

2πlR 2 exp(− β ⋅ E sf ,cyl ( x) ) ⋅ x ⋅ dx vacc ∫0

Eq. 16

where l is the length of the cylinder. Do et al.40 have shown that the Henry constant can become negative at high temperature if it is calculated over all the cell volume. A more suitable definition was proposed based only on the accessible pore volume. In this work we used this convention. The accessible pore radius (Racc) can be defined as:

Racc = R − σ sf

Eq. 17

where σsf is the solid-fluid collision diameter. By doing this, the integral is carried out in the region of the pore where the solid-fluid potential has an important contribution to the configurational integral. In other words, the region where adsorbed molecules and the solid atoms overlap, is neglected (note that the potential might still be slightly negative due to the contribution of the additional solid atom layers). Mathematically speaking, the upper bound of the integration in Eq. 15 and Eq. 16 is set to

xacc = 1 −

1 R*

Eq. 18

instead of 1.This does not introduce a significant source of error since for r > Racc Esf tends to infinity and therefore the Boltzmann factor goes to zero ( exp[ − E sf ( x ) β ] ≈ 0 ).


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K sph = K cyl =

3β 3 xacc 2β x 2 acc

xacc

∫ exp(− β ⋅ E

sf , sph

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( x) )⋅ x 2 ⋅ dx

Eq. 19

( x) )⋅ x ⋅ dx

Eq. 20

0

xacc

∫ exp(− β ⋅ E

sf ,cyl

0

Two approaches can be used to compute the integrals in Eq. 19 and Eq. 20 in order to obtain explicit expressions: either the integrand is approximated by a Taylor series expansion or a quadrature method is applied. In order to allow a Taylor expansion of the integrand, several simplifications are required, i.e. the repulsive part of the potential is neglected and only a single layer pore wall is considered. The exercise is carried out in the appendix for the case of a cylindrical pore. It presents the advantage of yielding an explicit expression for the Henry constant from which we can draw important qualitative conclusions. We will come back to the interpretation of the explicit in section 4.2.2. For practical purposes, the Taylor series expansion is not very suited, since the number of terms needed in the Taylor expansion increases rapidly as the pore diameter is increased. Quadrature methods such as the Riemann mid-point method41 are more precise and efficient for calculating the Henry constant so that we have used these methods. The Riemann midpoint method approximates the integral of a curve by a sum over a large number of segments (see supporting information for details). xacc

[

]

∫ exp − Esf ( x)β xdx = 0

xacc I

I −1

∑ exp[− E i =1

sf

( x(i )) β

]

Eq. 21

I is an integer defining the number of segments in which the function is divided. The value of the variable x(i) is given by x(i ) =

xacc  1  i +  I  2

Eq. 22


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In order to achieve a good numerical precision, the number of segments was set to I = 120. Applying the Riemann mid-point method to Eq. 19 and 20, yields the following final expressions for the Henry constant:

K sph =

3β 3 I ⋅ xacc

  mmax '     ∑ E sph ( R * + m∆*, x(i )*, ε sf , ρ sur *)  β  x(i ) 2 exp ∑     i =0     m =0

Eq. 23

K cyl =

2β 2 I ⋅ xacc

  mmax '     ∑ Ecyl ( R * + m∆*, x(i )*, ε sf , ρ sur *)  β  x(i ) exp ∑     l =0     m =0

Eq. 24

I −1

I −1

Eq. 23 and Eq. 24, express the Henry as a function of the microscopic descriptors of the solid: R*, ∆*, εsf, ρ*sur. These parameters take into account the size and shape of the pore, the density of the solid and the strength of its interaction with the adsorbate. Also it must be noted that all these parameters can be treated independently, so that we can find the adsorbent properties leading to an optimal Henry constant for a given adsorption process. We finally note that also the enthalpy of adsorption at zero coverage (∆H0) can be calculated from the analytical expression for the solid-fluid potential according to:

∆H 0 = N av

∫E

sf

(r ) exp(− Esf (r ) / kT )dr

v

∫ exp(− E

sf

(r ) / kT )dr

− RT

Eq. 25

v

where Nav is the Avogadro number. The heat of adsorption (q) is the negative value of the enthalpy of adsorption at zero coverage (q = -∆H0). Both integrals in Eq. 25 can be calculated using the Riemann mid-point integration method for cylindrical and spherical pores described previously. 3.5. Parameters of the solid-fluid and fluid-fluid LJ potentials The crossed solid-fluid LJ potential parameters can be obtained from the "pure fluid" LJ parameters σii and εii by means of the Lorentz-Berthelot combination rules: σ sf = (σ ss + σ ff ) / 2

Eq. 26

ε sf = ε ss ⋅ ε ff

Eq. 27 15

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In the following examples, all-silica zeolites were chosen as model adsorbents since in the behavior of such materials in adsorption is mainly governed by dispersion-repulsion forces, in accordance with the basic assumption of our solid-fluid potential. Since only the oxygen atoms are accessible on the surface of a zeolite, the dispersion-repulsion forces can be evaluated by integrating the silica contribution into the oxygen atoms.42,43 In this work the Lennard-Jones parameters for the oxygen atoms in all-silica zeolites were taken from the literature (εss = 93.03 K and σss = 3.00 Å).44 These parameters have been shown to be transferable. The choice of the LJ parameters and excluded volumes of the adsorbate molecules CO2 and CH445 is explained in detail in the supporting information. Note that the single LJ center models that are available in the literature for the CO2 molecule46-48 include the quadrupolequadrupole interaction in the dispersion forces, i.e. εff is artificially increased. This will lead to an overestimation of the solid-fluid potential between CO2 and a non-polar solid, which does not at all interact with the quadrupole moment of CO2. In order to correctly estimate the contribution of the dispersion forces in the Henry constant, we have developed a new spherical model of CO2 that dissociates dispersion forces and electrostatic contributions (see supporting information). This modified εff was used for calculating the solid-fluid interactions (i.e. it was used in Eq. 27), while the standard value of εff (incorporating the quadrupolequadrupole electrostatic forces) was kept for the fluid-fluid interactions. Table 1 summarizes the LJ parameters and excluded volumes of CO2 and CH4 that were used in this study. Table 1 Lennard-Jones fluid-fluid parameters and excluded volumes27 used in this study Adsorbate

σff, Å

εff, K

b, Å3/molecule

CH4

3.73

150

62

CO2

3.72a

203a

60

3.64b

232b

a

excluding the contribution of the quadrupole for the solid-fluid interactions

b

including the contribution of the quadrupole 16 ACS Paragon Plus Environment

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3.6. Summary of the input parameters of the descriptor-RSM model Summing up, the required input parameters of the descriptor-RSM model are: -

the Lennard-Jones parameters of the solid, εss and σss,

-

the Lennard-Jones parameters of the fluid, εff and σff,

-

the excluded volume b of the adsorbate,

-

the atomic volume density of the solid ρa (atoms per unit cell volume),

-

the atomic surface density of the solid ρsur (atoms per surface area),

-

the pore volume per mass of the solid VPore,

-

the pore geometry (sphere or cylinder),

-

the pore radius R.

The Lennard-Jones parameters and excluded volumes were discussed in the previous section. The atomic volume density ρa is easily calculated from the crystallographic structure (number of atoms in a unit cell divided by its volume). The surface density ρsur is determined by dividing the number of surface atoms in the unit cell by the Connolly surface area of a unit cell. The Connolly surface area can be calculated from the Atom Volumes & Surfaces model of Materials Studio. The RSM calculates a fractional loading of the pore volume θ. In order to convert this dimensionless loading into conventional mol/g units, the following equation is applied. n=

θ ⋅ VPore b

Eq. 28

where n is the adsorbed amount in mmol/g, b is the molar volume of the adsorbed phase at saturation expressed in cm3/mmol, i.e. the excluded volume expressed in other units. The pore volume may calculated theoretically from the crystal structure, but in this work we preferred to use experimentally measured pore volumes. 4. Results

4.1. Validation of the model by comparison with experimental data


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We start by comparing the theoretical Henry constants and isotherms of CH4 and CO2, calculated by the methods outlined above, with the experimental data of all-silica zeolites with FAU, LTA and MFI topology, i.e. the solids DAY, ITQ-29 and silicalite-1. The input parameters for these solids were determined as outlined above. In all-silica zeolites, only the oxygen atoms are in direct contact with the adsorbate molecules (the silicon atoms are hidden),43 therefore, only the oxygen atoms were taken into account in the calculation of the surface density. Once the parameters associated to the interaction potentials and the structural parameters of the solid (surface and volume density) are known, the equivalent pore geometry and pore radius that correctly describes the confinement of the real pore system of all-silica FAU, LTA and MFI has to be determined. In a separate work33 we have shown that the so-called pore shape index SI is able to indicate whether the real pore system most resembles a sphere (SI = 0) or a cylinder (SI = -1) or mainly consists of saddle points (SI = 1). The pore shape index can, therefore, be used to decide whether the choice of a cylindrical or a spherical pore model is more appropriate for our calculations. The pore shape index clearly identifies FAU and LTA as spherical structures while MFI is in majority assigned to a saddle-point surface (with SIs ranging from 0.2 to 1). Since no analytical expression for the solid-fluid potential in hyperbolic geometries (consisting of saddle points) is available, we have therefore tested both the spherical and the cylindrical pore model for the case of MFI. The accessible pore radius Racc of the "equivalent" sphere or cylinder was identified with the mean curvature of the real pore system, calculated by the method described in ref 33. Table 2 compiles the descriptors of the solid that are required for the model. In the case of all-silica zeolite A we encountered technical difficulties in determining the mean curvature of the pore system by the method described in ref 33. Therefore, the characteristic length of the


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all-silica zeolite A was taken from the literature using the largest sphere included in the zeolite framework.49

Table 2. Microscopic descriptors of the all-silica zeolites examined in this study Zeotype

Pore shape

Racc, Å

ρsur,b Oxygen/ Å2

∆, Å

VPore, ml/g 50-52

FAU

Spherical

6.88

0.17

3.32

0.34

MFI

Hyperbolic

2.24

0.17

3.00

0.16

LTA

Spherical

5.53a

0.15

3.28

0.34

a

Calculated by the largest included sphere. 49

b

Calculated using the Connolly 53 surface area. Using the values of Table 2, the Henry constants and zero coverage heats of adsorption of

CO2 and CH4 on the three all-silica zeolites were calculated by the model and compared with experimental values. In all the three cases, there is a good agreement between the experimental and the theoretical heats of adsorption of CH4 and CO2 at zero coverage. As can be seen in Table 3, the calculated values of q for CH4 and CO2 on silicalite-1 show that the spherical pore provides a better prediction of the experimental enthalpies. The same holds true for the Henry constants. This is not surprising since the range of the pore shape indices starts at 0.2, i.e. closer to sphere than to a cylinder. Therefore only the spherical representation of silicalite-1 will be used hereafter.

Table 3 Comparison between the heat of adsorption of CO2 and CH4 at zero coverage on all-silica zeolites predicted by Eq. 25 and experimental values.


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q CH4

q CO2

(kJ/mol)

(kJ/mol)

Theoretical

Experimental

Theoretical

Experimental

11.8

14.4 52

17.0

15.4 52

23.5

21 54-18.6

55

26.8

27 54-24.1 55

16.9

21 54-18.6 55

18.6

27 54-24.1 55

13.0

15 50

15.0

21 50

DAY Silicalite-1 (spherical) Silicalite-1 (cylindrical) ITQ-29

Table 4 Comparison between the experimental Henry constants of CO2 and CH4 on allsilica zeolites and the theoretical calculations. Solid

T, K

Kx103 of CH4

Kx103 of CO2

(molecule/bar*Å3)

(molecule/bar*Å3)

Theoretical Experimental Theoretical

Experimental

a

1.04

1.07 a

DAY

300

0.58

0.56

Silicalite-1

304

2.15

2.85 55

8.29

11.78 55

304

3.87

2.85

55

8.68

11.78 55

303

0.43

0.56 50

0.85

1.83 50

(spherical) Silicalite-1 (cylindrical) ITQ-29 a

Calculated from the experimental data at 300 K in references 52,56

b

Calculated from the experimental data at 304 K in reference 54 The theoretical Henry constants as given in Table 4 were used within the framework of the

RSM (Eq. 7) so as to obtain the single component adsorption isotherms of CH4 and CO2. Figure 3 compares the theoretically predicted isotherms with experimental data. Note that the isotherms were calculated without using a single adjustable parameter. In the case of DAY and all-silica zeolite A the agreement between the theoretical RSM adsorption isotherms and the experimental data is excellent. Only, only the isotherm of CO2 on silicalite-1 is not 20 ACS Paragon Plus Environment

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perfectly reproduced. We attribute this discrepancy to a bad estimation of the pore volume available for adsorption. Silicalite-1 has the smallest pore size of all the structures investigated in this study. As a consequence, the volume available for adsorption depends on the size and shape of the adsorbate. Hence, the N2 pore volume is not fully transferable to CO2. This shows that our method has limits when looking at small pore sizes.

8

4

6

3

n [mmol/g]

n [mmol/g]

4 2 0

2 1 0

0.1

1

p [bar]

10

100

0.1

1

10

100

p [bar]

(a)

(b) 8 6 n [mmol/g]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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4 2 0 0.1

1

p [bar]

10

100

(c) Figure 3 Comparison between experimental adsorption isotherms and the prediction of the RSM (solid line) on (a) DAY, CH4 (diamonds) and CO2 (squares) at 300 K from reference 52, (b) silicalite-1, CH4 (circles at 298 from 7 and diamonds at 304 K from reference 55) and CO2, circles at 304 K from reference 54 squares at 303 K from reference 57and (c) all-silica zeolite A, CO2 (squares) and CH4 (diamonds) at 303 K reference 50.


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4.2. Parameter sensitivity study: qualitative behavior of the model 4.2.1. Comparison of the confinement effect in spherical and cylindrical geometries As it was shown in the previous section, in cases where the assignment of an equivalent simplified geometry is ambiguous (as for MFI), the choice between spherical or cylindrical geometry has an impact on the resulting confinement. The fact that the mean curvature of a sphere is two times larger than the mean curvature of a cylinder with the same diameter perfectly illustrates the stronger confinement effect exhibited by the sphere. To gain a better understanding of the confinement effect in spherical and cylindrical pores, the adsorption enthalpy of CO2 at zero coverage was calculated for both geometries using equation Eq. 25. In order to consider only geometric effects, the density of the solid and the product ερsur were held constant. Therefore, the pore diameter was the only parameter changing the heat of adsorption. Figure 4 shows that for small pores the heat of adsorption in cylindrical pores is higher than the one found in spherical pores. As expected, for large pore sizes both geometries tend to the same heat of adsorption. This is because in both cases, the adsorption potential approaches the potential energy of a molecule interacting with a flat surface when R goes to infinity i.e. to zero curvature. To quantitatively assess the enhancement of confinement in spherical pores an interpolation of the curves in Figure 4a was carried out. Then, the ratio of the spherical and cylindrical pore radius (Rsph/Rcyl) was taken for points corresponding to the same heat of adsorption (see Figure 4b). The Rsph/Rcyl ratio is always in the range 1-1.4 and shows a maximum at intermediate values of ∆H0. This maximum can be explained as follows: for large pore radius both geometries tend to the same behavior (flat geometry). As the pore radius is reduced the attractive part in spherical pores increases faster than in the cylindrical pore. Nevertheless, for very small pore radii the repulsive forces also increases more strongly in a spherical pore than


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in a cylindrical pore. Therefore the trade-off between the attractive and repulsive forces gives rise to the maximum. In conclusion, we can state that the difference in the effective confinement between a sphere and cylinder is not 2 (as expected from the ratio of mean curvatures), but rather between 1.2 and 1.4 in the range of relevant pore sizes.

0

5

R* 10

2

15

20

0

-10

1.5 Rsph/Rcyl

-5 ∆H [kJ/mol]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1

-15

0.5

-20 0

-25

-25

-20

-15 -10 ∆H [kJ/mol]

-5

0

Figure 4 (a). ∆H0 of CO2 at 303 K vs. reduced pore radius and (b) ∆H0 of CO2 at 303 K vs. Rsph/Rcyl ratio. Spherical pore (diamonds), cylindrical (squares), SED* = 0.72 (full symbols), SED*= 0.94 (open symbols) and ∆* = 0.88.

4.2.2. Relationship between the solid's properties, the Henry constant and the heat of adsorption Having validated the geometrical pore model by comparison with experimental data, we can now use the model to evaluate (semi-)quantitatively the effect of the solid's different microscopic descriptors on the Henry constant. According to section 3.2., the parameters that govern the Henry constant are the surface energy density ρ*surεsf, the pore radius R, the Lennard-Jones diameter σsf and the density of the solid ∆. The density of solid is not truly an independent parameter, but is related to the pore radius. The larger the pore radius, the smaller is the density of the solid. In order to take the relationship between density and pore radius


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into account, we have used an available empirical correlation between the radius and the density for zeolites49: ρ s = −0 .1182 R acc + 2 .089 g / cm 3

Eq. 29

where Racc must be expressed in Å to obtain the density in g/cm3. Note that the offset of 2.089 g/cm3 is close to the skeletal density of dense SiO2. The mass density ρs was transformed into an atomic density by using the molar mass of SiO2, i.e.

ρa = ρs ⋅

3 ⋅ N av M SiO 2

Eq. 30

CO2 was used as a model adsorbate for the calculations. Since the surface energy density (SED* = 0.8 – 0.95 for CO2) and the Lennard-Jones diameter (σsf = 3.36 for CO2) are rather constant from one zeolite to another, the only remaining parameter was the pore radius. Figure 5 shows the evolution of the Henry constant of CO2 at 303 K for cylindrical and spherical pores as a function of the pore diameter D* = 2R*. For cylindrical pores K decreases as the pore diameter is increased (see Eq. 36 and Eq. 37). For the spherical geometry the value of K goes through a maximum. We had already mentioned before that the increase of repulsive forces with decreasing pore diameter is more pronounced in spherical geometry. This leads to the decrease of the Henry constant at D* < 2.4. Also, it must be noted that the Henry constant decreases faster with the pore diameter in the cylindrical pore than in the spherical pore. This is due to its lower confinement effect. We further investigated the impact of the strength of adsorbate-adsorbent interaction on the Henry constant by varying the surface energy density. Increasing the SED, as would be the case for a more strongly interacting molecule, shifts the curve upwards, i.e. to higher Henry constants, but the shape of the curve is unchanged (the maximum in spherical geometry is at the same position). As expected from the approximate expression of the Henry constant that is given in the appendix (Eq. 36 and Eq. 37), there is a linear relationship between the logarithm of the Henry constant and the value of SED. 24 ACS Paragon Plus Environment

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100

Kx103 [molecules/bar*Å3]

1000

SED* SED* SED* SED* SED*

100000

SED* = 0.65 SED* = 0.80 SED* = 0.94 SED* = 1.07 SED* = 1.20

10000

10 1

3

3

Kx10 [molecules/bar*Å ]

100000

0.1

10000 1000 100

= = = = =

0.65 0.80 0.94 1.07 1.20

10 1 0.1

0.01

0.01 2

2.5

3

3.5

4

4.5

5

5.5

2

2.5

3

3.5

D*

4

4.5

5

D*

(a)

(b)

Figure 5. Henry constant of CO2 at 303 K as a function of the reduced pore diameter (D* = 2R/σsf) for different values of SED. (a) cylindrical pore and (b) spherical pore. Figure 6 shows the heat of adsorption of CO2 at zero coverage as a function of the mean curvature of the pore, i.e. 1/2Racc for cylinders and 1/Racc for spheres. Here Racc was calculated from the pore radius by using Eq. 17. The curves in Figure 6 can be approximated by a straight line for curvature < 0.5 Å-2, which is the trend that we empirically found in ref 33. The slope of this line is controlled by the SED (product ρsur εsf).


50 40

SED* = SED* = SED* = SED* = SED* =

60 50 q [kJ/mol]

60

q [kJ/mol]

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30 20 10

40

0.65 0.80 0.94 1.07 1.20

30 20 10

0

0

0

0.2

0.4

0.6

0.8

1

0

0.2

-1

mean curvature [Å ]

(a)

0.4

0.6

0.8

1

-1

mean curvature [Å ]

(b)

Figure 6. Isosteric heat of adsorption of CO2 as a function of the pore mean curvature. (a) cylindrical pore (b) spherical. The empirical relationship pore radius adsorptiondensity was used.


5.5

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4.3. Application of the model: working capacity for CO2 separation by PSA on non-polar adsorbents In the previous section we have analyzed the relationship between the Henry constant and the microscopic descriptors of the solid. Small pore sizes and large surface energy densities lead to high Henry constants. However, our objective in adsorption and separation processes is not necessarily to maximize the Henry constant. Adsorption and separation processes like PSA (Pressure Swing Adsorption) are cyclic operations where an adsorption step at high pressure is alternated with a desorption step at low pressure. High Henry constants lead to very steep adsorption isotherms that are favorable for adsorption, but unfavorable for desorption. A good compromise between both steps is desired. The parameter that needs to be maximized is ∆n, the working capacity of the adsorbent.58 The working capacity is defined as the difference between the adsorbed amounts of CO2 during the adsorption and the desorption steps: ∆ nCO 2 = nCO 2 , p 2 − nCO 2 , p1

Eq. 31

where nCO2,p1 and nCO2,p1 are respectively the CO2 uptakes at the pressure p1 (low pressure) and p2 (high pressure). The RSM does not provide an absolute, but a fractional working capacity, which can, however, be used as a surrogate: ∆ θ = θ CO 2 , p 2 − θ CO 2 , p1

Eq. 32

The adsorption pressure p2 was fixed to 5 bar and the desorption pressure p1 was fixed to 1 bar. The PSA cycle was considered to be isothermal at 303 K. Figure 7 shows the fractional working capacity obtained for spherical and cylindrical pores as a function of the pore diameter.



0.35 0.30 0.25 ∆θ

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0.20


0.35 0.30 0.25 ∆θ

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0.20

0.15

0.15

0.10

0.10

0.05

0.05 0.00

0.00 6

8

10

12

14

16

18

20

6

8

10

12

14

16

18

20

D [Å]

D [Å]

(a)

(b)

Figure 7. Fractional working capacity of CO2 vs. pore diameter for a PSA process cycle operating between an adsorption pressure of 5 bar and a desorption pressure of 1 bar (a) cylindrical pores and (b) spherical pores. For both cylindrical and spherical pores, the fractional working capacity goes through a maximum when varying the pore diameter. This maximum is shifted to larger pore diameters in the case of the spherical pores. The ratio between the optimal pore diameters of the sphere and the cylinder is approximately 1.2, which corresponds to the difference in confinement effects between a sphere and a cylinder identified in the previous section. When examining the impact of the SEDs we can see that higher values of this parameter also lead to higher optimal pore diameters. The relationship between SED and the optimal diameter is almost linear. The impact of both effects (pore geometry and SED) can be rationalized as follows: for a given pressure range, there is a given value of the Henry constant that will maximize ∆θ (within the mathematical framework of the RSM). In spherical geometry, the pore size leading to this optimal Henry constant is larger than in cylindrical geometry because of the higher confinement effect found in the case of the sphere. Increasing the SED increases the Henry constant, which must be compensated by increasing the pore diameter in order to attain the optimal value of the Henry constant. The fact that there is a given value of the Henry constant that will optimize the fractional working capacity also explains why a second maximum of ∆θ appears at very small pore 27 ACS Paragon Plus Environment

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diameters in the case of the spherical geometry. Since the Henry constant goes through a maximum as a function of the pore diameter (see Figure 5b), the optimal value of K can be reached at two different pore diameters, one on the left, the other on the right side of the maximum. The reasoning above was based on the fractional working capacity, i.e. we have identified the values of the microscopic descriptors that will lead to the most efficient use of the pore volume in a PSA cycle. However, from a practical point of view the main objective in the design of a PSA process is to minimize the volume of the adsorption vessel and, hence, the volume of the adsorbent. In order to do that, the working capacity must be expressed in moles of adsorbate per volume of adsorbent. In order to convert the fractional loading into a working capacity per volume of adsorbent, ∆θ has to be multiplied by the porosity φ of the adsorbent, which is, as we recall, also dependent on the pore radius. The porosity can be evaluated from the ratio between the mass density of the porous solid and the skeletal density of the framework, i.e. the skeletal density of SiO2 in the case of all-silica zeolites.

ϕ = 1−

ρs ρ sk

Eq. 33

By using the empirical relationship between density and pore size that was established for zeolites, we obtain:

ϕ=

0.1182 ⋅ Racc 0.1182 ⋅ (D / 2 − σ sf ) = 2.089 2.089

Eq. 34

Figure 8 shows the product ∆θ *φ as a function of the pore diameter for spherical and cylindrical pores.


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SED* = SED* = SED* = SED* = SED* =

0.080

0.65 0.80 0.94 1.07 1.20

0.08

0.040 0.020


0.06 ∆θφ

0.060 ∆θφ

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0.04 0.02

0.000

0.00 6

8

10

12

14

16

18

20

6

8

D [Å]

10

12

14

16

18

20

D [Å]

(a)

(b)

Figure 8. CO2 working capacity per adsorbent volume between 1 and 5 bar vs. pore diameter for (a) cylindrical pore (b) spherical pore. Since the porosity linearly increases with the pore diameter, the optimal working capacity is slightly shifted to larger pore diameters, compared to Figure 7. The second maximum at small pore diameters that we found in Figure 8 for the spherical geometry has disappeared in Figure 9, because small pore diameters are associated with a low porosity and, hence, a low volumetric adsorption capacity. Figure 8 also demonstrates that larger pore diameters may potentially lead to larger optimal working capacities since a larger fraction of the adsorbent volume is available for adsorption. However, the larger pore volume is only efficiently exploited when the SED is sufficiently high, i.e. for very strong adsorbent-adsorbate interactions. The typical values of the SED* (product ρ*sur εsf /kT) for the adsorption of CO2 in all-silica zeolites are in the range between 0.80 and 0.95. Therefore, for this kind of solids, the optimal working capacity is obtained for pore diameters between 9.3 and 10.2 Å in the case of a cylinder and between 11.4 and 13.5 Å in the case of a sphere. The position of the optimum is not significantly modified when the adsorption step is changed from 5 to 10 bar (see supporting information). Note that these optimal pore diameters are within the range found in large pore zeolite structures. 29 ACS Paragon Plus Environment

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5. Conclusions Via the Ruthven Statistical Model we have succeeded in establishing a link between the microscopic descriptors of an adsorbent/adsorbate system and its macroscopic adsorption properties. The inputs of the original RSM are a Henry constant describing the solid-fluid interaction, a fluid-fluid interaction potential, the excluded volumes of the adsorbates and the volume available for adsorption, i.e. it already contains some important microscopic descriptors of the system. The challenge of our approach was to link the Henry constant to some key properties of the adsorbate and the adsorbent. We have succeeded in establishing this link by calculating the Henry constant theoretically from analytical expressions of the Lennard-Jones solid-fluid potential in simplified pore geometries, i.e. spheres and cylinders. The parameters needed for calculating the solid fluid potential, i.e. the descriptors of the adsorbate/adsorbent system are: the (accessible) pore radius, the surface density of the solid (atoms per surface area), the volume density of the solid (atoms per unit cell volume) and the depth of the solid-fluid potential well. The link between simple and real pore geometries, i.e. between spheres and cylinders and arbitrary shapes, is established via the mean curvature of the real solid. Having evaluated the mean curvature of the accessible surface of the real solid,33 one can calculate the characteristic length of the pore system by assuming cylindrical or spherical geometry. This characteristic length is then identified with the accessible pore radius and used in the descriptor-RSM model. Thanks to this approach, it is possible to reproduce experimental adsorption data (Henry constants, heats of adsorption and even the full adsorption isotherm) of non-polar adsorbents in the absence of any single adjustable parameter. Having thereby validated the descriptor-RSM model, we have used it to analyze the effect of the microscopic descriptors, in particular the pore size and the surface energy density SED, ,on the macroscopic adsorption properties. It was found that the Henry constant exponentially 30 ACS Paragon Plus Environment

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increases with the value of SED and decreases when the confinement effect decreases (larger pore diameters). We can further use the RSM model to calculate the value of the Henry constant that will maximize the working capacity in a cyclic adsorption process and then identify the corresponding couple of pore diameter and SED. To illustrate the methodology, we have considered as an example a PSA cycle for CO2 adsorption operating in the range of pressures between 5-1 bar (adsorption-desorption). In the range of SEDs that is encountered in CO2 adsorption on all-silica zeolites, the optimal pore radius is between 9.3 and 10.2 Å for the case of a cylinder and between 11.4 and 13.5 Å for the case of a sphere. If the SED increases, i.e. the solid-fluid interactions are stronger than in the previous example, the optimal pore diameter shifts to larger values. The methods developed in this paper open the door for the intelligent design of new adsorbents for a given cyclic adsorption process. Yet, we also have to recall the limits of descriptor-RSM. The solid-fluid potential only considers dispersion-repulsion forces, therefore the calculations are only valid for non-polar adsorbents. Furthermore, the assumption that the adsorbates have spherical shape becomes problematic if the pore size approaches the molecule diameter. Since it is well-known that polar solids are the best materials for CO2 separations, since they have a high CO2/CH4 selectivity,10,59,60 we will extend our approach to electrostatic interactions in future work and attempt to integrate them into the RSM model. Appendix We demonstrate an explicit calculation of the Henry constant in a cylindrical pore by solving the integral Eq. 20 via a Taylor expansion. In order to simplify the Taylor expansion, the repulsive part of the solid-fluid potential is neglected and, second only a single layer pore


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wall is considered. With these simplifications, the attractive part of the solid-fluid potential in a cylindrical pore Esf,cyl (Eq. 10 and Eq. 12) is approximated by :  9 2 9 4 1 + x + x  3π ⋅ SED  4 64  E sf ,cyl ( x ) = ⋅ *4 4 R (1 − x) (1 + x) 4

Eq. 35

This integrand x•exp(-Esf,cyl(x)•β) of Eq. 16 is in turn Taylor-expanded in the center of the pore, i.e. the function is approximated at x = 0. Integration of the Taylor polynomial then yields K cyl ≈

2 2 R gasT ⋅ x acc

 1 A 2 25 A  e A  1225 625 2  4 6 e x + e A ⋅ x + A+ A  ⋅ xacc + ...  acc acc  16 6  64 32  2 

Eq. 36 where

A=

3π ⋅ SED * R *4

Eq. 37

The same procedure can be applied to the case of the spherical pore and yields an expression of a similar functional form. For both geometries, the logarithm of the Henry constant is in first approximation proportional to the surface energy density and inversely proportional to the fourth power of the reduced pore radius.

Abbreviations EoS

Equation of State

LJ

Lennard-Jones

PSA

Pressure Swing Adsorption

RSM

Ruthven Statistical Model

SED

Surface Energy Density


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