Effects of Enhanced Flexibility and Pore Size Distribution on

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Effects of Enhanced Flexibility and Pore Size Distribution on Adsorption-Induced Deformation of Mesoporous Materials Andrei Kolesnikov, Nikolaj Georgi, Yury A. Budkov, Jens Möllmer, Jorg Hofmann, Juergen Adolphs, and Roger Gläser Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00591 • Publication Date (Web): 24 May 2018 Downloaded from http://pubs.acs.org on May 24, 2018

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Eects of Enhanced Flexibility and Pore Size Distribution on Adsorption-Induced Deformation of Mesoporous Materials ∗,†,‡

Kolesnikov A.L.,

Georgi N.,

¶

§,k

Budkov Yu. A.,

Adolphs J.,

†Institut

and Gläser R.

†

Hofmann J.,

†,⊥

für Nichtklassische Chemie e.V., Permoserstr. 15, 04318 Leipzig, Germany

‡Porotec

GmbH, Niederhofheimer Str. 55A, 65719 Hofheim am Taunus, Germany

¶GMBU, §G.

‡

†

Möllmer J.,

Erich-Neuÿ-Weg 5, 06120 Halle (Saale), Germany

A. Krestov Institute of Solution Chemistry of the Russian Academy of Sciences, Akademicheskaia 1, Ivanovo, Russia

k

Tikhonov Moscow Institute of Electronics and Mathematics, School of Applied

Mathematics, National Research University Higher School of Economics, 34 Tallinskaya Ulitsa, 123458, Moscow, Russia

⊥Institut

für Technische Chemie, Universität Leipzig, Leipzig, Germany

E-mail: [email protected]

Abstract We present a new model of adsorption-induced deformation of mesoporous solids. The model is based on a simplied version of Local Density Functional Theory in the framework of solvation free energy. Instead of density, which is treated as constant here, we used the lm thickness and pore radius as order parameters. This allows us to obtain a self-consistent system of equations describing simultaneously the processes of gas 1

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adsorption and adsorbent deformation, as well as conditions for capillary condensation and evaporation. In the limit of innitely rigid pore walls, when the lm becomes several monolayers thick the model reduces to the well-known Derjaguin-Broekhode Boer (DBdB) theory for pores with cylindrical geometry. We have investigated the eects of enhanced exibility of the solid as well as the inuence of the pore size distribution on the adsorption/deformation process. The formulation of the theory allows to determine the average pore size and its width from desorption branch of the strain isotherm only. The model reproduces the non-monotonic behavior of the strain isotherm at low relative pressure. Furthermore we discuss the eect of rigidity of the adsorbent on the pore size distribution, showing qualitatively dierent results of the adsorption isotherms for rigid and highly exible materials. In particular the shift of evaporation pressure to lower values and the absence of a limiting value of the loading at high relative pressure. We discuss as well the results of the theory with respect to experimental data obtained from the literature.

Keywords Strain isotherm, stress, water adsorption, DBdB model, capillary condensation, nitrogen adsorption, Vycor glass

Introduction Adsorption of gases in a porous medium is caused by adsorbent-adsorptive interactions. While it is common to regard changes of the adsorptive density at the pore wall upon changes of thermodynamic state, the changes of the adsorbent during gas adsorption have only recently attracted attention. The mechanical response of the adsorbent to gas adsorption, referring to as adsorption-induced deformation 14 is caused by the stress due to a competition between adsorptive-adsorptive and adsorbent-adsorptive interactions. These interactions

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may result in either swelling or shrinkage of the porous body. The type of deformation depends on adsorbent and absorptive properties as well as on experimental conditions. This is exemplied by the dierent forms of strain isotherms that have been observed in experiments: in the most common case, adsorption is accompanied by expansion exhibiting either a concave or a linear form; 1,4,5 monotonic contraction 68 during water adsorption on silica and heptane adsorption on mesoporous silicon; non-monotonic behavior, contraction at low pressures followed by expansion with increasing pressure, during heptane adsorption on porous silica. 2 While all the above examples refer to deformation of mesoporous materials, the discussion of adsorption-induced deformation and the shapes of strain isotherms has been limited to pressure range preceding capillary condensation. Typical magnitudes of adsorption-induced deformation lie between fractions and tens of percents. Strains exceeding several percents are common for materials with enhanced exibility such as polymers, 9 aerogels 10,11 and MOFs. 12 In this respect it is worth noting that, although, for polymers and solid materials the typical strains could be of comparable magnitude, the mechanisms and the nature of adsorption-induced deformation are dierent. In recent years vivid interest in the adsorption-induced deformation has triggered a number of experimental and theoretical investigations. From the theoretical point of view adsorption-induced deformation has been studied employing a thermodynamic approach, 5,1216 density functional theory 1721 and computer simulations. 2224 Ravikovitch et al., using nonlocal density functional theory (NLDFT), developed a theory linking adsorption and strain for materials with spherical pores. 25 The authors of the paper 25 were rst to show that adsorption stress is equal to the solvation pressure and can be obtained from the grand thermodynamic potential of the uid in the pore. Also, they have shown that their model is capable to describe, almost quantitatively, non-monotonic adsorption-induced deformation of zeolites and other microporous materials. Based on thermodynamics, the authors of 14 addressed adsorption-induced deformation of a porous material with cylindrical pore geometry, using the Derjaguin-Broekho-de Boer (DBdB) theory with Gibbs adsorption 3


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isotherm. For rigid materials the deformation of the whole sample can be described by the eect of adsorption within the pore that increases the "solvation pressure". 14,25,26 The pore width, remains unaltered within this model, which makes it possible to exclude the variation of the pore width upon adsorption. This approach was further extended to account the eect on pore anisotropy. 27 In the publication 12 it has been shown that the thermodynamic approach from 14 gave similar results compared to the quenched solid density functional (QSDFT) theory. In the paper 20 the authors have addressed the eect of pore size distribution (PSD) on the strain isotherms of Argon in a microporous activated carbon. In recently published papers, 21,28 it has also been shown that information from strain experiments could be exploited to determine the PSD. In the range of micropores this is a particular advantage, since in contrast to the adsorption isotherms, the strain isotherms are more sensitive to the pore width. Grosman et al. 13 developed a thermodynamic approach, taking into account the adsorption-induced deformation via the elastic energy of the solid matrix. Ustinov et al. 18 have studied, within the framework of NLDFT, the eects of coupling deformation and adsorption on heat of adsorption, solvation pressure and density distribution of the uid. They found that the quantities are signicantly aected by the deformation of the adsorbent. Diao et al. 22 demonstrated the inuence of packing and temperature eects on solvation pressure and strain isotherms. Vandamme et al. 16 have studied coal deformation subject to carbon dioxide adsorption by means of a modied poromechanics approach. Despite signicant progress in understanding of adsorption-induced deformation and the wide range of studied eects that deformation of a porous material during sorption causes, only few works exist that describe the coupling between deformation and adsorption within a thermodynamic approach taking self-consistently into account the variation of the pore width upon adsorption. 13,18,22,29,30 The aim of the present work is to develop such a theory, that would self-consistently treat adsorption-induced deformation taking into account pore width variation and altered solid-uid interactions upon adsorption. This in turn allows to estimate the eect of adsorption on the PSD and to assess how rigid a material must 4


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be for adsorption to have a negligible eect on the PSD. Furthermore the present theory is able to describe non-monotonous deformation prior to adsorption as well as the pressure region of capillary condensation/evaporation and above. This allows to extend the range of applicability of adsorption-induced deformation thermodynamics based theories to materials with enhanced exibility (more than 1% of volume deformation).

Model We consider a process of adsorption in the pore as growth of a liquid-like lm on the surface assuming complete wetting of the surface. The liquid-like lm is represented as a liquid with bulk properties bounded by a solid surface from the one side and with vapor from the another side. For the sake of compact notation, we refer in the following to the "liquid-like lm" as "liquid lm". The sample is considered as a porous solid material with a porosity φ, which will be dened below, and bulk modulus K . In order to describe the thermodynamic properties of the whole sample, we use the solvation Gibbs free energy of the adsorbent as a function of order parameters that will be specied below. In what follows we formulate the whole equation and then specify each term separately:

∆Gsolv = Fsample + Fads + P Vsample − µNads ,

(1)

where P and µ is the pressure and chemical potential of the vapor, Vsample is the volume of the whole sample and Nads is the number of adsorbed molecules in the sample. The method is based on minimization of solvation Gibbs free energy and has been successfully applied to describe the deformation of metal-organic frameworks (MOFs) 31 as well as the behavior of a polymer coil in solution. 3236 The rst term in (1) is the Helmholtz free energy of the sample that consists of two terms:

Fsample = Fref + Fdef , 5


(2)

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where Fref is an energy of the solid matrix at reference conditions and Fdef is the energy of elastic deformation. The reference state of the solid is chosen to be a "dry" state with a negligible initial stress. The second term in Eq.(2) takes into account the contribution from the elastic energy of deformation of the whole solid sample (including the change of surface area) and can be expressed by the Hook's law. 37 The second term in Eq. (1)describes the contribution to the thermodynamic potential from the adsorption process. Liquid vapor surface tensions are approximated by the values from non-curved interface γlv . Clearly, this is a simplication, since the properties of thin liquid lm are not identical to the bulk liquid (see below). Notwithstanding the limitations, similar simplifying assumptions have been employed to describe adsorption, 3841 such that we can expect at least qualitative agreement with experimental results. Thus, the lm free energy can be written as a sum of the bulk free energy of the liquid lm, the contribution from the interactions of the lm molecules with the solid, the surface free energy and the free energy of the gas phase in the pore:

Fads = Flv + Ff ilm + Fext + Fg ,

(3)

where Flv is the free energy of the 'liquid-vapor' interface and Ff ilm = µ0 Nads − P0 Vads is the free energy of the bulk liquid with pressure P0 and chemical potential µ0 taken at the saturation conditions, which amounts to an in-compressible liquid. 38 The third term R Fext = ρ(r)Vext (r)dr is the energy of solid-liquid interactions, which are taken into account only for the liquid lm, and ρ(r) is the density of uid within this lm. It is further assumed that the solid-gas interactions are absent and therefore the gas phase in the pore is the same as in the reservoir; Fg = µNg − P Vg is the free energy contribution from the gaseous phase. In order to obtain the correct behavior of the thermodynamic potential and adsorption stress for the limiting case of the lm thickness approaching zero, the dependence of liquid-vapor free energy on the lm thickness must be taken into account. In particular the following

6


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conditions have to be satised: When the lm thickness h tends to zero, the surface free energy also must tend to zero, while when h tends to innity, the surface free energy must approach the limit of Slv γlv , where Slv is the liquid vapor interface with arbitrary geometry. Based on these conditions, we prescribe the surface free energy of the liquid-vapor interface the following functional form Flv = Slv γlv (1 − e−h/λ ), which describes the exponential reduction of surface free energy with decay length λ, which is at the order of the monolayer thickness1 . Combining the written above, equation (1) takes the form:

∆Gsolv (Nads , V ) = Fref (V0 ) + Fdef + Flv + µ0 Nads − P0 Vads + µNg − P Vg + Fext + P Vsample − µ(Nads + Ng ),

(4)

where V is the volume of the whole sample at specied pressure P/P0 . The volume of the sample can be written as a sum of volumes of the adsorbed phase, the gas phase in the pores and the volume of the solid, the latter is considered to remain unaltered:

Vsample = Vads + Vg + Vs .

(5)

0 , Also, the porosity of the sample in the reference state can be dened as φ = Vp0 /Vsample

where Vp0 is the volume of the pores. Hence, (4) could be expressed as:

∆Gsolv (Nads , V ) = Fref (V0 ) + Fdef + Vs P + Flv

(6)

+ (µ0 − µ)Nads + (P − P0 )Vads + Fext , 1 The

motivation for this approach is related to the fact that the surface free energy of the liquid-vapor interface is ill-dened for lm thickness of molecular dimensions. Notwithstanding the particular functional form, the correct qualitative behavior of the surface free energy Flv is enforced by the limiting behavior at h → 0 and h → ∞. The particular functional form merely serves the purpose to interpolate the function between the two extremes.

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Thus, we have obtained the thermodynamic potential for a porous sample. Thriving to obtain information on the level of individual pores, we will make the assumption of "independent pores" and the assumption of "independent domains", meaning that a broad pore size distribution consists of a combination of domains with a xed pore size. The porous material is assumed to be homogenous and isotropic and therefore should be described by an eective bulk modulus and average value of mesoporous porosity. Also, the resulting stress is considered to arise from the radial stress in the cylindrical pore. The assumption made above in line with the simplication made in the generalized adsorption equation upon which the calculation of the PSD from adsorption isotherms and an adsorption model for an individual pore rests. We treat the pores as the perfect cylinders with dierent radii and the adsorption layer as a cylindrical shell with inner radius R − h and outer radius R. The density in the adsorption layer is ρl , thus the adsorbed amount in one pore is:

Nads = ρl Vads = ρl πL[R2 − (R − h)2 ],

(7)

the liquid-gas surface area of adsorbed lm is:

Slv = 2πL(R − h).

(8)

Without loss of generality the pores with equal pore width can be characterized by the same length, because the only condition on the length is L 2R. Thus, the total solvation free energy of upon the above assumptions takes the following form:

{k}

∆Gsolv (h

,R

{k}

(V k − V0k )2 k k φM + Slv γlv (1 − e−h /λ ) ) = Fref (V0 ) + Vs P + k 2 V0 k k k k −kB T ln(P/P0 )Nads + (P − P0 )Vads + Fext , X 1

(9)

where the sum over k refers to the summation over all domains with radii Rk and lm 8


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thicknesses hk . The elastic energy is expressed through the volume of pores, average porosity and eective bulk modulus of the whole sample. Thus, the whole potential can be expressed as a sum of the contribution from independent pores in an elastic medium with average bulk modulus M and porosity φ. The external potential is assumed to have an exponential form:

Vext (z) = we−z/λ

(10)

where w is the energetic parameter, and λ is the decay length, which is also chosen in the order of monolayer thickness. The spatial coordinate z denotes the distance from the pore surface to the edge of liquid lm on it. Moreover, for small changes of the pore radius R, the energetic parameter w can be approximated by a series expansion:

∂w (R − R0 ) kB T = −(w0 − κ(R − R0 ))kB T, w = w(R)kB T ≈ − |w(R0 )| − ∂R

(11)

where w0 = |w(R0 )|, κ = ∂w/∂R > 0 and kB - is the Boltzmann constant. Thus, within the present model, the solid-uid interactions in the pore are allowed to vary with the pore wall deformations. Positive κ reects the decrease of the solid-uid interactions upon expansion. The motivation behind the pore radius dependent solid-uid interactions rests upon the assumption that the strength of surface - uid interactions is primarily determined by the number of adsorption sites on the surface. While the number of adsorption sites during deformation remains xed, 15 the expansion causes a decrease in adsorption sites density and in turn to a reduction of the adsorbent - adsorbate interactions. Summarizing, the contribution from uid - surface interactions can be expressed as:

Z Fext =

ρ(r)Vext (r)dr = −2πLρl kB T (w0 − κ(R − R0 ))λ (λ − R) e−h/λ − 1 + he−h/λ , (12)

where we used above discussed approximation of liquid density prole. Worth noting, that 9


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the use of uid-solid adsorption potential will result in the appearance of an exponentially decaying disjoining pressure in the liquid lm on the surface and in the pore lled with liquid. In the case of negligible κ it will be equal to recently used form of disjoining pressure. 5 Thus, the use of the exponential form of the interaction potential is motivated by the widespread use in the adsorption community. 5,4244

Model: Adsorption branch

The equilibrium values of pore radius and lm thickness can be found by minimizing (9) with respect to R and h:

∂∆Gsolv (h, R) = 0, ∂h ∂∆Gsolv (h, R) = 0. ∂R

(13)

For the sake of simplicity we have neglected the index k , since for all pores the equations are identical. This results in a following set of equations, that need to be solved simultaneously:

R2 − R02 1 Rg T Rg T φM = h ln(P/P0 ) − h(P − P0 ) + |w(R)|λ(1 − e−h/λ ) (14) 2 R0 R vm vm κλRg T −h/λ −h/λ −h/λ − γlv (1 − e )− (λ − R) e − 1 + he Rvm Rg T |w(R)|Rg T −h/λ γlv (1 − e−h/λ ) γlv −h/λ − ln(P/P0 ) = e + − e + P0 − P, (15) vm vm R−h λ where Rg - is the gas constant. The condensation is considered as a loss of stability of a metastable liquid lm on the pore surface with the condition for the lm stability:

∂ 2 ∆Gsolv (h, R) ∂ 2 ∆Gsolv (h, R) ∂ 2 ∆Gsolv (h, R) ≥ 0, − ∂h2 ∂h2 ∂R2

10


∂ 2 ∆Gsolv (h, R) ∂h∂R

2 ≥ 0.

(16)

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Model: Desorption branch

The thermodynamic potential for the pore after condensation can be obtained by a substitution of Rk for hk in the Eq.(4). The so obtained equations reads as:

(V k − V0k )2 2 V0k k k k k −kB T ln(P/P0 )Nads + (P − P0 )Vads + Fext ,

{k} ∆Gdes ) solv (R

= Fref (V0 ) + Vs P +

X 1

φM

(17)

Minimizing the thermodynamic potential with respect to the radius, the equilibrium radius of the pore can be found:

∂∆Gdes solv = 0, ∂R

(18)

This in turn leads to the following expression for the equilibrium radius of the pore:

R2 − R02 1 Rg T Rg T −R/λ φM = R ln(P/P0 ) − R(P − P0 ) + |w(R)|λ(1 − e ) (19) R02 R vm vm Rg T κλ − λ e−R/λ − 1 + R . Rvm Turning to the desorption process, the evaporation is considered as an equilibrium transition between the lled pore and the pore with a liquid lm on the surface. 45 Worth noting, that pores with dierent dry radii considered separately, due to approximation of independent domains. Hence, the thermodynamic potential of the pore with lm (9) should be equal to the thermodynamic potential of the lled pore (17): ∗ des ∆Gads solv (h , Rads ) = ∆Gsolv (Rdes ).

(20)

The equation denes the evaporation condition, where Rads and Rdes are the pore radii of the partially lled and the completely lled pore, respectively and h∗ is the equilibrium lm 11


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thickness. In the limit of a completely rigid sample or for a material with a suciently high bulk modulus, the latter equation reduces to the condition, similar to the well-known Broekho-de Boer equation: 46

2γlv (1 − e−h Rg T ln(P/P0 ) = − vm R − h∗

∗ /λ

)

+

2|w(R)|Rg T vm

RR h∗

dt(R − t)e−t/λ

(R − h∗ )2

+ P0 − P.

(21)

For |w(R)| approaching zero and h∗ approaching a value of several monolayers, the above equation reduces to the Kelvin equation. 47,48 The latter happens because the exponent in the rst term and the second term are vanished in that limit.

Model: Pore size distribution

In order to calculate PSD we will use the generalized adsorption isotherm equation (GAI) in matrix notation: (22)

N = Ax,

where N is the vector representing the experimental isotherm, A is the matrix,whose elements

Aij represent the loading in the pores and x is the sought function representing the PSD. The units of the PSD derive from the normalization condition of the matrix A. The index i enumerates the pressure and j - the pore radius, which is the result of the Eqs.(15,20) and depends on the pressure. In order to solve the equation (22), the pore width in the reference state (Wj,ref ) and the pressure are discretized. The matrix entries then take the form:

Aij =

   ρl π[Rj (pi )2 − (Rj (pi ) − h)2 ],

pi ≤ p∗

(23)

∗

  ρl πRj (pi )2 ,

pi > p

where p∗ is the pressure of the capillary condensation/evaporation. The adsorbed amount in the pore is normalized to the pore length, which is assumed to remain unaltered. Conse12


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quently the physical meaning of the vector x is the length of a pore with a specied width in the reference state. This in turn allows us to obtain the pore volume distribution (P V D) of the sample in the reference state (Wj,ref ):

P V D(Wj,ref ) = bT ◦ x,

(24)

and the total volume of the pores in the sample in the reference state:

V0 = bx,

(25)

2 where the vector b denotes a volume per length: vj = πWj,ref /4. The volume change of the

pores in the sample then reads as:

V = Gx,

(26)

where G is a matrix with entries Gij = πRj (pi )2 . Thus, the swelling of the whole sample is expressed as:

S(pi ) =

δVSample = φ(Vi − V0 )/V0 . 0 VSample

(27)

Adsorption-induced deformation in experiments is commonly measured recording a relative length change of the sample. In order to compare the results of the theory with experimental values, volumetric deformations were converted to linear deformations via

δV /V0 ≈ 3δl/l0 , which is valid for small deformations δl/l0 1. Eq. (22) has been solved using the NNLS (non negative least squares) algorithm.

Results and discussions The here presented theory contains several unknown parameters: sample porosity (φ), energetic parameter (w0 ), decay length (λ) and eective bulk modulus (M ). In order to compare the theoretical results with experimental ndings we have chosen the decay length and the

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porosity as input parameters. The decay length has been identied with the width of a monolayer. The value for the porosity has been correlated with the data from literature. Since the deformation is more sensitive to the change of interaction potential than the adsorption in our model, the energetic parameter w0 , and the eective bulk modulus M were used to t the strain isotherm. The eective bulk modulus was obtained from the slope of the desorption branch of strain isotherm. Thus, all required parameters could be obtained from the strain isotherm only. The pore size distribution is obtained from adsorption isotherm or, alternatively, from the desorption branch of strain isotherm. Table 1 shows the parameters used in all forthcoming calculations. Table 1: The list of parameters that were used in calculations for dierent uids, namely: temperature, molar volume at saturation pressure, monolayer thickness, surface tension of vapor - liquid at interface. The parameters for water refer to adsorption on Vycor glass and nitrogen on a hypothetical material. Fluid Water Nitrogen

T /K 292 77.4

vm / 18.0 34.6

cm3 mol

hm / Å γlv / 2.4 72.9 3.54 8.88

mN m

Fig. 1a shows dierent strain isotherms at dierent values of energy w0 and κ parameters. The isotherms are plotted only prior to capillary condensation, i.e. for the adsorption branch. Variation of both parameters results in dierent shapes of the strain isotherms observed in experiments on mesoporous materials and obtained in theoretical studies. 2,48,15 For strong interactions w0 = 8.5 and negligible κ the strain isotherm has a concave shape. Within the whole pressure range increase of the pressure leads to an expansion of the material. For reduced w0 , the expansion is proportional to the pressure and the isotherm shows a linear shape. For weak interactions e.g. for small values of w0 = 5 and a non zero κ, the isotherms exhibit a non-monotonic behavior - at low P/P0 the material contracts, at higher pressures the material expands. Higher κ values lead to a monotonic contraction of the sample within the whole pressure range. The described behavior originates from two competing tendencies swelling due to soliduid interactions (reduction of surface energy) and the shrinkage due to the reduction of 14


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(a)

(b)

Figure 1: (a) - Strain isotherm of water for dierent w and κ values demonstrate qualitatively dierent behavior of the adsorption-induced deformation - concave, linear, non-monotonic, convex strain isotherm shapes. Worth noting, that the gap between curves calculated with dierent values of w0 is the result of the particular choice of the adsorbent-adsorptive interaction potential and the assumption of liquid layer with constant density on the surface. It predicts the non-innite minimal value of chemical potential from which the adsorption starts. However, we believe, that the latter does not change the results of the model qualitatively. (b) - Normalized dry critical radii of evaporation for three dierent relative pressures p/p0 = 0.4, 0.6, 0.8 as a function of eective bulk modulus M. For the sake of comparison for all curves the critical radius was chosen from a material with bulk modulus M = 20 Gpa. The calculations were performed with the set of parameters corresponding to Nitrogen at 77 K, w0 = 13 and κ = 0. adsorbent - adsorbate interactions during swelling. In absence of a coupling between expansion of the pore and surface-uid interactions (e.g. for negligible κ in Eq. (11) ) the adsorption-induced deformation is governed by the surface-uid interactions that always tend to expand the pores. For weak surface-uid interactions the expansion linearly depends on the pressure. At the other extreme for non-negligible κ the deformation is governed by a weakening of the surface-uid interactions that arises from a decrease of the density of adsorption sites on the surface. Thus for non-negligible κ the last term in Eq. (11) dominates, leading to a contraction of the material irrespective of the lm thickness. For intermediate cases at low pressures the contribution due to a decrease in adsorption sites density leads to a contraction, at higher pressures the contribution of surface-uid interactions dominates 15


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leading to an expansion with a minimum in between. It is worth noting, that the account of the surface-uid interactions w0 as a function of pore radius is reminiscent of the eect of surface stress. The inuence of surface stress has recently been described by the authors of 15 in the framework of a BET-like model, taking into account the expansion of solid surface via the BET constant C . The model has been shown to qualitatively reproduce the experimentally observed deformation curves prior to condensation. Fig. 1b shows the dependence of dry critical radii (radius of dry pore for which evaporation happens at certain relative pressure and bulk modulus) on the eective bulk modulus M. The values of critical radii are calculated with the system of equations: (13), (18) and (20) for three dierent values of relative pressure 0.4, 0.6 and 0.8. One can see that decrease of bulk modulus causes a decrease of dry critical radius too. That could interpreted as follows: for sti materials (M 1 GPa) swelling only mildly aects the pore radius and a as sequence the dry critical radius of evaporation remains approximately the same. However, for softer materials, the swelling could reach values of several percentages and thus the dierence between dry radius and actual radius of swollen pore could be pronounced. The latter in turn leads to a decrease of the critical radii for evaporation. The inuence of the eective bulk modulus on the strain isotherm, amounts to changing the magnitude of the deformation, thereby shifting the strain isotherm vertically, whereas the critical radius changes the location of the capillary condensation/evaporation. In the remaining part of the section we will consider three dierent examples of adsorptioninduced deformation. Focusing on the eect of deformation on the PSD of hypothetical mesoporous materials within the whole pressure range, we will investigate the inuence of materials' elasticity and address the question to which extent the exibility of the material aects the PSD. Subsequently we present the results of theory when applied to experimental data on adsorption-induced deformation for water adsorption on Vycor glass.

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Hypothetic material with low elastic constant

In this section we rst assess the consequences of the adsorption-induced deformation model on a hypothetical material with very low eective bulk modulus M = 20 M P a, porosity

φ = 0.9, w0 = 8 and κ = 1nm−1 and parameters which should correspond to the Nitrogen adsorption. Figs. 2a and 2b show theoretical adsorption and strain isotherms in the region of relative pressure above 0.8, respectively. Parameters were chosen in order to qualitatively reproduce the deformation behavior reminiscent of aerogels 10 at high relative pressures. In contrast to the experimental data, due to the Hook's law approximation, the model can predict only reversible deformations and certainly underestimates the magnitude of deformation. Also, the same parameters were used for rigid isotherm calculation, i.e. eective bulk modulus M 1 GP a. We used Gaussian pore length distribution with average pore width of 16 nm for the both theoretical adsorption isotherms: 2 max − 12 ( x−x σ ) , e g(x) = √ 2πσ

(28)

where max determine the pore volume per mass of adsorbent, x is the average pore width and σ is the standard deviation. While the isotherm of almost rigid material reproduces the typical type H1 hysteresis, the isotherm with low bulk modulus shows deviations. In particular, branch of the isotherm is shifted to low relative pressures and there is no saturation value of the amount adsorbed, instead the adsorbed amount monotonically increases up to ≈ 1 relative pressure. Both deviations are due to the fact that the pores of the sample are not rigid and therefore there volume changed throughout the pressure range. Thus, the shift of the evaporation pressure occurs because of rapid decrease of the capillary radius during desorption. Also, the absence of saturation loading occurs because of the monotonic increase of pore radius in the regime of completely lled pores. There is only minor shift of the adsorption branch of the isotherm, due to the low magnitude of deformation during capillary condensation.

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(a)

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(b)

Figure 2: (a) - Theoretical adsorption isotherms corresponding to dierent eective bulk moduli (M = 20 M P a and M −→ ∞ ) of the sample, while all other parameters were xed. (b) - Theoretical strain isotherm corresponding to the case of very low bulk modulus M = 20 M P a and κ = 1.25 nm−1 . Water on Vycor glass

In the rst example we consider water adsorption on Vycor glass at 292 K. The analysis is based on experimental data from adsorption and desorption isotherms as well as from the strain isotherms published in the work. 4 Fitting the model simultaneously to adsorption isotherm and the strain isotherm the following parameters were obtained within the framework of the present model: M ≈ 13 GPa and w ≈ 10.6, the porosity of the sample was taken as φ = 0.3. Fig. 3a shows the adsorption and desorption isotherms as obtained from the t of the model to experimental data alongside with experimental data. 4 Fig. 3b shows the PSD as obtained from the adsorption and desorption isotherms. The theoretical isotherms follow the experimental data, but the minor overestimation prior capillary condensation is observed. The calculated average pore diameter is ∼ 7.5 nm. The pore size distribution obtained from the adsorption branch is slightly broader, which could be due to the type H2 hysteresis typical for pore-blocking eects. 49 Fig. 3c shows the experimental data of the strain isotherms corresponding to adsorption

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(a)

(b)

(c)

(d)

Figure 3: (a) - Adsorption isotherm of water on the Vycor glass at 292 K: symbols denote experimental data from. 4 Lines show the results of the present theory for adsorption and desorption branches. (b) - Pore size distribution as obtained from adsorption and desorption branches of the isotherm. The line denoted by Gaussian PSD shows the pore size distribution as obtained from the strain isotherm of water on the Vycor glass at 292 K. (c) Comparison between experimental data and theoretical description of strain isotherm with the pore size distributions from adsorption isotherm of water on the Vycor glass at 292 K. (d) - Comparison between experimental data and the theoretical model, for water adsorptioninduced deformation of Vycor glass at 292 K. The pore size distribution has been prescribed a Gaussian functional form (28). The theoretical deformation curve was obtained from strain isotherm data. and desorption branches. The strain isotherms were obtained using the PSD's from Fig.3b from adsorption and desorption branches, while all the other parameters remained xed.

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Minor shifts between the deformation curves in Fig.3c arise from the PSDs as obtained from adsorption and desorption branches of the isotherm in Fig.3a. While the use of PSDs from adsorption and desorption branches result in a slight shift below the capillary condensation, the slope of the two deformation curves coincide above capillary condensation. A marked feature of the strain isotherm is the large but continuous deformation during desorption in the region of capillary evaporation - at P/P0 ∼ 0.7. Taking into account a distribution of pore width, results in a continuous deformation curve, that would otherwise, in the case of a single pore width, result in discontinuity during capillary evaporation. It is also worth mentioning that the theoretical deformation curves deviate at low pressures from experimental deformation curves. This is to be expected, since the underlying assumption of the model is the presence of a liquid lm for the adsorption branch. After having obtained the PSD from adsorption and strain isotherms, we will attempt to obtain the PSDs from deformation data only. Since strain isotherm record a relative change of the sample volume, information on absolute volume is in principle not available in this representation. We will for this reason assume the PSD to have the functional form of a Gauss distribution (28) with the prefactor deliberately set to the value, which reproduce similar pore volume, as we obtained the adsorption isotherm. This in turn implies that the position and the width can be obtained from relative deformation data, while the pore volume can not be determined. To compare the PSD from strain isotherm only to the PSDs obtained from adsorption isotherm and strain isotherm we use the not exactly the same value of parameter

w0 , as previously obtained from a t to the adsorption isotherm. The new value of w0 is ≈ 10, and was obtained from the best t of strain isotherm. However, small changes of w0 have only a minor eect on the PSD. In this respect it is also worth noting, that the parameter w0 shifts the PSD, while the width of PSD is primarily determined by the slope of the strain isotherm. Fig. 3d shows the experimental strain isotherm alongside with model strain isotherm for the parameters described above. The pore size distribution is presented in Fig. 3b. Comparing the results of the model for the strain isotherm obtained from deformation data only with 20


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the experimental data in Fig.3d, the model curve closely matches the experimental data points. The model curve in Fig.3d provides similarly good t to experimental data as the model curves in Fig.3c. The close coincidence between theoretical and experimental curves in Fig.3d and Fig.3c indicate that the PSD can be determined from strain isotherm only with comparable accuracy to the case where in addition the adsorption isotherm has been used. The PSD obtained from strain isotherm only and the prescribed functional form of a Gaussian is shown in Fig.3b. The maxima and the width of the Gaussian PSD in Fig.3b are in qualitatively good agreement with the PSDs from adsorption and desorption isotherms.

Conclusions In this work we have suggested a theory describing the adsorption-induced deformation based on the construction of the generalized thermodynamic potential. 3236 We have obtained selfconsistent system of nonlinear equations for the strain and adsorption isotherms, as well as conditions for evaporation and lm stability. We have showed that in the case when pore walls are innitely rigid, and the lm thickness reaches the value of several monolayers this model reduces to the well-known DBdB 44,46,50 theory for pores with cylindrical geometry. We have taken into account the pose size distribution of the sample and obtained that it leads to better agreement with experimental data in the region of evaporation. We also have obtained that for ordered mesoporous materials the average value of the PSD and its width could be obtained from strain isotherm only. The model has been tested on example from the literature 4 - water sorption at ambient temperature on Vycor glass. The obtained results are in good agreement with previously proposed theories. 14,15 The account of the changes of eective adsorbent-adsorbate potential of interaction during pore deformation allows to obtain qualitative description of dierent experimentally observed strain isotherms, particularly with non-monotonic behavior at low relative pressure. Also, the model qualitatively describes the features of nitrogen adsorption isotherms on aerogels, 10 in particular the shift

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of evaporation pressure to lower values and absence of a limiting value of the loading at high relative pressure. We also would like to stress the main limitations of the model, which follow from the assumptions we have made: negligible zero stress (i.e. the stress in the material at dry conditions), elastic deformation of the sample, cylindrical geometry of the pores and the existence of liquid-like lm on the surface. Despite the simplications, the results of our theory are in good agreement with experimental data. To overcome the present limitations we plan to extend the present theory, which shall be the subject of forthcoming publications.

Acknowledgements We gratefully acknowledge the nancial support from Bundesministerium für Wirtschaft und Energie (BMWi) as part of the program "Zentrales Innovationsprogramm Mittelstand (AiFZIM)" (ZF4186801GM5, ZF4129902GM5 und ZF4186502GM5). Also, Y. Budkov thank Russian Federal Program, grant no RFMEFI61618X0097 for support.

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(3) Eriksson, J. C. Thermodynamics of surface phase systems. Surface Science 1969, 14, 221246. (4) Amberg, C. H.; McIntosh, R. A study of adsorption hysteresis by means of length changes of a rod of porous glass. Canadian Journal of Chemistry 1952, 30, 10121032.

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adsorption-desorption process: A thermodynamic approach. Physical Review B - Condensed Matter and Materials Physics

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(23) Schoen, M.; Paris, O.; Günther, G.; Müter, D.; Prass, J.; Fratzl, P. Pore-lattice deformations in ordered mesoporous matrices: experimental studies and theoretical analysis. Physical Chemistry Chemical Physics

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(25) Ravikovitch, P. I.; Neimark, A. V. Density functional theory model of adsorption deformation. Langmuir : the ACS journal of surfaces and colloids 2006, 22, 1086410868. (26) Evans, R.; Marini Bettolo Marconi, U. Phase equilibria and solvation forces for uids conned between parallel walls. The Journal of chemical physics 1987, 86, 71387148. (27) Balzer, C.; Waag, A. M.; Gehret, S.; Reichenauer, G.; Putz, F.; Hüsing, N.; Paris, O.; Bernstein, N.; Gor, G. Y.; Neimark, A. V. Adsorption-Induced Deformation of Hierarchically Structured Mesoporous Silica - Eect of Pore-Level Anisotropy. Langmuir

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(30) Günther, G.; Prass, J.; Paris, O.; Schoen, M. Novel insights into nanopore deformation caused by capillary condensation. Physical Review Letters 2008, 101, 2023. (31) Coudert, F.-X.; Fuchs, A. H.; Neimark, A. V. Adsorption deformation of microporous composites. Dalton Trans. 2016, 45, 41364140.

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(32) Budkov, Y. A.; Kolesnikov, A. Statistical description of co-nonsolvency suppression at high pressures. Soft Matter 2017, (33) Budkov, Y. A.; Kolesnikov, A. L.; Georgi, N.; Kiselev, M. G. A exible polymer chain in a critical solvent: Coil or globule? EPL (Europhysics Letters) 2015, 109, 36005. (34) Budkov, Y. A.; Kolesnikov, A. L.; Kalikin, N. N.; Kiselev, M. G. A statistical theory of coil-to-globule-to-coil transition of a polymer chain in a mixture of good solvents. EPL (Europhysics Letters)

2016, 114, 46004.

(35) Kolesnikov, A. L.; Budkov, Y. A.; Basharova, E. A.; Kiselev, M. G. Statistical theory of polarizable target compound impregnation into a polymer coil under the inuence of an electric eld. Soft Matter 2017, 13, 43634369. (36) Budkov, Y. A.; Kiselev, M. Flory-type theories of polymer chains under dierent external stimuli. Journal of Physics: Condensed Matter 2017, 30, 043001. (37) Landau, L. D.; Lifshitz, E. Theory of Elasticity, vol. 7. Course of Theoretical Physics

1986, 3, 109. (38) Ustinov, E.; Do, D.; Jaroniec, M. Equilibrium adsorption in cylindrical mesopores: a modied Broekho and de Boer theory versus density functional theory. The Journal of Physical Chemistry B

2005, 109, 19471958.

(39) Kowalczyk, P.; Jaroniec, M.; Terzyk, A. P.; Kaneko, K.; Do, D. D. Improvement of the Derjaguin- Broekho- de Boer Theory for Capillary Condensation/Evaporation of Nitrogen in Mesoporous Systems and Its Implications for Pore Size Analysis of MCM-41 Silicas and Related Materials. Langmuir 2005, 21, 18271833. (40) Georgi, N.; Kolesnikov, A.; Uhlig, H.; Möllmer, J.; Rückriem, M.; Schreiber, A.; Adolphs, J.; Enke, D.; Gläser, R. Characterization of Porous Silica Materials with

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Water at Ambient Conditions. Calculating the Pore Size Distribution from the Excess Surface Work Disjoining Pressure Model. Chemie Ingenieur Technik 2017, 89, 16791685. (41) Kolesnikov, A. L.; Uhlig, H.; Möllmer, J.; Adolphs, J.; Budkov, Y. A.; Georgi, N.; Enke, D.; Gläser, R. Pore size distribution of MCM-41-type silica materials from pseudomorphic transformation - A minimal input data approach based on excess surface work. Microporous and Mesoporous Materials 2017, 240, 169177. (42) Ravikovitch, P. I.; Vishnyakov, A.; Neimark, A. V.; Ribeiro Carrott, M. M. L.; Russo, P. A.; Carrott, P. J. Characterization of Micro-Mesoporous Materials from Nitrogen and Toluene Adsorption:Experiment and Modeling. Langmuir 2006, 22, 513516, PMID: 16401094. (43) Schlangen, L. J. M.; Koopal, L. K.; Stuart, M. A. C.; Lyklema, J.; Robin, M.; Toulhoat, H. Thin Hydrocarbon and Water Films on Bare and Methylated Silica: Vapor Adsorption, Wettability, Adhesion, and Surface Forces. Langmuir 1995, 11, 17011710. (44) Kolesnikov, A. L.; Uhlig, H.; Möllmer, J.; Adolphs, J.; Budkov, Y. A.; Georgi, N.; Enke, D.; Gläser, R. Pore size distribution of MCM-41-type silica materials from pseudomorphic transformation-A minimal input data approach based on excess surface work. Microporous and Mesoporous Materials 2017, 240, 169177. (45) Ravikovitch, P.; Domhnaill, S. Ó.; Neimark, A.; Schüth, F.; Unger, K. Capillary hysteresis in nanopores: theoretical and experimental studies of nitrogen adsorption on MCM-41. Langmuir 1995, 11, 47654772. (46) Broekho, J.; De Boer, J. Studies on pore systems in catalysts: IX. Calculation of pore distributions from the adsorption branch of nitrogen sorption isotherms in the case of open cylindrical pores A. Fundamental equations. Journal of Catalysis 1967, 9, 814.

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(47) Chen, Y.; Wetzel, T.; Aranovich, G. L.; Donohue, M. D. Generalization of Kelvin's equation for compressible liquids in nanoconnement. Journal of colloid and interface science

2006, 300, 4551.

(48) Melrose, J. C. Model calculations for capillary condensation. AIChE Journal 1966, 12, 986994. (49) Lowell, S.; Shields, J. E.; Thomas, M. A.; Thommes, M. Characterization of porous solids and powders: surface area, pore size and density ;

Springer Science & Business

Media, 2012; Vol. 16. (50) Neimark, A. V.; Ravikovitch, P. I.; Vishnyakov, A. Bridging scales from molecular simulations to classical thermodynamics: density functional theory of capillary condensation in nanopores. Journal of Physics: Condensed Matter 2003, 15, 347.

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Pressure For Table of Contents Only

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0 ,9 5

p /p 0

= 0 .4

p /p 0

= 0 .6

p /p 0

= 0 .8

0 ,9 0

0

c r it

(M ) / R

0

c r it

(2 0 G P a ) / -

1 ,0 0

R

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0 ,8 5 1

1 0

M

/ G P a


Effects of Enhanced Flexibility and Pore Size Distribution on

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