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Deformation of microporous carbons during N2, Ar, and CO2 adsorption: Insight from the density functional theory Christian Balzer, Richard Tyler Cimino, Gennady Y. Gor, Alexander V. Neimark, and Gudrun Reichenauer Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b02036 • Publication Date (Web): 15 Jul 2016 Downloaded from http://pubs.acs.org on July 16, 2016

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Deformation of microporous carbons during N2, Ar, and CO2 adsorption: Insight from the density functional theory Christian Balzer*,†, Richard Cimino‡, Gennady Y. Gor§, Alexander V. Neimark*,‡ and Gudrun Reichenauer*,† †

Bavarian Center for Applied Energy Research (ZAE Bayern), Magdalene-Schoch-Straße 3, 97074, Wuerzburg, Germany ‡ Rutgers, The State University of New Jersey University, Department of Chemical and Biochemical Engineering, 98 Brett Road, Piscataway, NJ 08854, United States § NRC Research Associate, Resident at Center for Materials Physics and Technology, Naval Research Laboratory, Washington, DC 20375, USA Keywords: carbon, adsorption, density functional theory, adsorption-induced deformation, microporous materials, carbon dioxide, argon, nitrogen Abstract: Using the nonlocal density functional theory, we investigate adsorption of N2 (77 K), Ar (77 K) and CO2 (273 K) and respective adsorption-induced deformation of microporous carbons. We show that the smallest micropores comparable in size and even smaller than the nominal molecular diameter of the adsorbate contribute significantly to the development of the adsorption stress. While pores of approximately the nominal adsorbate diameter exhibit no adsorption stress regardless of their filling level, the smaller pores cause expansive adsorption stresses up to almost 4 GPa. Accounting for this effect, we determined the pore-size distribution of a synthetic microporous carbon by simultaneous fitting its experimental CO2 adsorption isotherm (273 K) and corresponding adsorption-induced strain measured by in-situ dilatometry. Based on the pore-size distribution and the elastic modulus fitted from CO2 data, we predicted the sample’s strain isotherms during N2 and Ar adsorption (77 K), which were found to be in reasonable agreement with respective experimental data. The comparison of calculations and experimental results suggests that adsorption-induced deformation caused by micropores is not limited to the low relative pressures typically associated with the micropore filling, but is effective over the whole relative pressure range up to the saturation.

1. Introduction Adsorption-induced deformation of nanoporous materials is a structural change occurring in porous solids during adsorption of gases, vapors or ions. It is an intrinsic part of the adsorption process that has been experimentally investigated for nearly a century.1, 2 The observed volumetric strains of the majority of porous materials are below 1 %, with the prominent exceptions of low density mesoporous aerogels3 and metal organic frameworks,4 which were shown to exhibit strains exceeding 10 vol% during gas adsorption. As a consequence adsorption-induced deformation is neglected in most scientific and technical considerations. However, even in cases where observable strains of adsorbents are small, effects related to adsorption-induced deformation may be of importance. For example the dramatic reduction of permeability in natural coal during CO2 1 ACS Paragon Plus Environment

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sequestration was attributed to porosity changes resulting from adsorption.5 Given that adsorptioninduced stresses (in particular for slit-shaped micropores) are predicted to amount up to hundreds of megapascal6-8 furthermore the question arises whether stress of such magnitude can inflict damage to the adsorbent. Contrary to this, adsorption-induced deformation also offers opportunities for the characterization of nanoporous materials: measurements of adsorption-induced strain contain information about the mechanical properties of the pore matrix, which is often not directly accessible by other experimental techniques.9, 10 Additionally, it was recently shown that the adsorption stress isotherms measured by in-situ dilatometry provide useful information for the micropore size analysis, complementary to the commonly used methods based solely on the analysis of adsorption isotherms.10 Several concepts were put forward to describe adsorption-induced deformation quantitatively, but so far there is no universal theory applicable for all kinds of porous structures, rather a set of various approaches for different types of materials, see e.g. refs 11-13 for mesoporous solids, refs 7, 14-19 for microporous solids and ref 20 for metal organic frameworks. In this work we focus on the origin of adsorption-induced deformation of microporous carbons. Typically microporous materials, such as carbons or zeolites, exhibit a non-monotonic adsorption-induced deformation: they contract at low stages of micropore filling and cross over to expansion as adsorption progresses.21-32 The extent of observed contraction and expansion depends on the structure of the adsorbent,27, 30 the adsorptive applied23, 24, 31, 33-36 and the temperature.2, 28, 32, 37 The first model for adsorption-induced deformation of microporous materials was proposed by Bangham, who considered the energy change of the adsorbate-adsorbent interface to be the sole source of stress within the porous structure.38 This concept was later implemented into the framework of Biot’s poromechanics by the use of the Gibbs adsorption equation.17, 18, 39 However, the fundamental issue of this approach is that adsorption by common understanding leads to a reduction of the interfacial energy and thereby only expansion should be observed – in clear contradiction with experimental findings. Here, we apply the thermodynamic model proposed by Ravikovitch and Neimark16 employing the concept of adsorption stress, or solvation pressure. Within this model, the adsorption-induced deformation of an individual micropore was interpreted as arising from a competition between the short-ranged repulsive and the long-ranged attractive interaction of fluid molecules and the solid: for the initial stages of micropore filling the attractive interactions of adsorbed molecules with the pore walls of the solid dominate, causing the pore to contract. For proceeding adsorption, the density of adsorbed molecules builds up within the pore and the repulsive interactions eventually translate into an expansive adsorption stress. Computational studies on carbon micropores revealed a pronounced non-monotonic correlation between adsorption stress and micropore size depending upon the ratio of the pore size and the molecular diameter of the adsorbate.7, 18, 19, 40-43 Due to this correlation, the entire pore size distribution (PSD) of a carbon has to be taken into account in order to understand its adsorption-induced deformation.7 In the following, we extend and test the scope of the adsorption stress model of adsorptioninduced deformation in two ways: (i) utilizing nonlocal density functional theory (NLDFT)44 we investigate the adsorption stress in carbon micropores over a wide range of pores sizes, including the smallest micropores equal and smaller in size than the nominal molecular diameter of the adsorbing gas, which are commonly considered inaccessible for adsorption; (ii) we compare our computational results to experimental adsorption-induced strain data31 for the adsorption of CO2 (273 K), N2 (77 K) and Ar (77 K) on a sample of microporous carbon. While deformation induced in micropores by CO2 2 ACS Paragon Plus Environment

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adsorption was already investigated,10, 17, 43, 45 a comparison between experimental and theoretical strains for the adsorption of N2 and Ar at cryogenic temperatures has not been reported so far. In particular, we show that one set of structural parameters (pore size distribution (PSD) and elastic modulus) can reasonably predict adsorption-induced deformation for all three adsorbates. The paper is structured as follows: in section 2, we outline our approach including the principle of NLDFT calculations, the molecular models involved, and the derivation of adsorption and stress isotherms. In section 3, we analyze the computational results in correlation with the experimental data, with a special focus on micropores smaller than the nominal molecular diameter of the adsorbate. Conclusions are given in section 4. Throughout the paper pores are classified according to IUPAC recommendations: pores smaller than 2 nm are micropores, pores between 2 and 50 nm are mesopores, pores larger than 50 nm are macropores and all pores smaller than 100 nm a referred to as nanopores.46

2. Methods and computational details 2.1 NLDFT theory of adsorption NLDFT for the description of adsorption phenomena was introduced in refs 47, 48. The basis of NLDFT calculations is the formulation of the grand potential of the adsorbate as a functional of the number density () profile of the adsorbate in the pore: () = () +

() + d ()( () − )

() = d ()ln(

!

() − 1)

(2)

(1)

Here is the position vector, is the intrinsic Helmholtz free energy of the ideal gas,

is the excess free energy due to fluid-fluid interactions, the chemical potential, is the external potential imposed by the adsorbent, is the Boltzmann constant, is the absolute temperature,

and

= #2$% /ℎ( is the thermal de Broglie wavelength of an adsorbate molecule.

The minimization of the grand potential with respect to the density profile leads to the EulerLagrange equation () =

)!

*+, -−

.

() + − ()/ .()

(3)

which is solved at given temperature and chemical potential for the equilibrium adsorbate density profile () and the corresponding grand potential. 2.2 Molecular models

The excess free energy

of the fluid phase is separated into fluid-fluid attraction and hard sphere repulsion 12 contributions:

() = () + 12 ()

(4)

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1

() = 4 dd5 ()(5 )6 (| − ′|) 2

(5)

The attractive potential for the fluid-fluid interaction 6 is modeled according to the Weeks-Chandler-Andersen (WCA) perturbation scheme:49 >−?@@ < C( D 6 (:) = 4? FGE@@ H − GE@@ H I = @@ : : < ;0

C

: < :B = 2D E@@ :B < : < :

(6)

: > :

Here ?@@ and E@@ are parameters for depth and range of the interaction potential, respectively. :B describes the minimum of the potential and : is the cutoff distance of the fluid-fluid interaction. ?@@ and E@@ were adapted to correctly describe physical properties of the unconfined fluid, e.g. liquid surface tension and liquid-vapor equilibrium (Table 1).50

For the hard sphere repulsion free energy 12 , we employed the RLST version44 of the fundamental measure theory of Rosenfeld,51 which was used previously in ref 52:

12 () =

Y (O) (:) =

1 NC N( − PQR ∙ PQT N(! (1 − 3U ( + 2U ! )V (7) d M−NO ln(1 − N! ) + + 1 − N! 24$(1 − N! )(

.([ − :) , 4$[ (

NX () = d′ (′)Y (X) ( − ′)

Y (C) (:) =

.([ − :) , 4$[

(8)

Y (() (:) = .([ − :),

.([ − :) ^(QR) () = .([ − :), ^(QT) () = : : 4$[

Y (!) (:) = ]([ − :),

(9)

Here, NX () are weighted densities derived by utilizing the scalar weighting functions Y(X) and the vectorial weighting functions ^(X) ; additionally U = |P`( ⁄N( |.In Equation 9 .(:) is the Dirac delta function, ](:) the Heaviside function and [ = b12 ⁄2 the hard sphere radius of the adsorbate molecule with b12 being the respective hard sphere diameter. The carbon-fluid interaction potential inside a slit-shaped micropore is given by the overlapping potentials of the opposing walls, which are modeled by the well-known 10-4-3 Steele potential @ :53 @ (c) =

(c) = @ (c) + @ (dee − c)

2 E@ CO ( 2$2 ?@ E@ ∆F G H 5

c

(10)

g E@ E@ g −G H − I c 3∆(0.61∆ + c)!

(11)

Here dee is the distance of the centers of carbon atoms in opposing walls, 2 = 0.114 Å-³ is the solid atom density of graphite and ∆ = 3.35 Å is the interlayer spacing of graphite. ?@ and E@ are parameters for depth and range of the interaction potential, which were adapted to correctly describe the adsorption on nonporous graphite (Table 1).50 The parameters of intermolecular 4 ACS Paragon Plus Environment

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interactions used are summarized in Table 1. All calculations were performed with the in-house software52 adopted for single component adsorption in an external field.

Table 1: Summary of parameters applied in NLDFT calculations for the adsorption of N2, Ar and CO2.a

ijj /kl mjj iqj /kl mqj nop r sstu [Å] [K] [g/mol] [K] [Å] [K] [Å] N2 94.45 3.575 3.575 53.22 3.494 77.4 28.013 Ar 118.05 3.305 3.39 55.0 3.35 77.4 39.948 CO2 235.9 3.454 3.495 81.5 3.43 273.15 44.01 a Here ?@@ and E@@ are the parameters of the attractive fluid-fluid potential, b12 is the molecular hard sphere diameter, ?@ and E@ are the parameters of the solid-fluid potential,50 is the temperature and %Bv is the molar weight of the adsorbate.54 The cutoff distance for fluid-fluid interaction was 5E@@ , the cutoff distance for solid-fluid interaction 10E@ .

2.3 Adsorption isotherms and fitting of experimental adsorption data To compare the computational results with experimental adsorption data, the amount adsorbed in moles per unit pore area, wxyz{| , was calculated by integration of the adsorbate number density distribution (Eq. 3) for given pore width d and relative gas pressure ,/,O = exp(/ ): wxyz{| (,/,O , d) =

1 (c, ,/,O , d)dc 2w O

(12)

Here w is the Avogadro constant. The internal pore width, d = dee − Eee , is defined corresponding to the distance between the opposing pore walls, where Eee = 3.4 Å is the effective diameter of a carbon atom. To obtain PSDs from experimental adsorption isotherms we followed the common procedure of inverting the integral adsorption equation

w (,/,O ) = wxyz{| (,/,O , d) ∙ @@ (d)dd O

(13)

by the Tikhonov regularization method55 implemented as the quick non-negative least square method in a previous work;50 here w is the amount adsorbed by the sample and @@ is the differential pore area distribution. We present PSDs as cumulative and differential pore volume distributions, B and @@ , which are calculated respectively, as @@ (d) = d/2 ∙ @@ (d)

B (d) = @@ (d′)dd′ O

(14)

(15)

The NLDFT calculations used in this work were performed for internal pore widths d from 2 to 20 Å in steps of 0.1 Å for the adsorption of N2 at 77 K, Ar at 77 K and CO2 at 273 K. Notably this 5 ACS Paragon Plus Environment

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range includes pore widths smaller than the nominal diameters of the different adsorbates. The covered relative pressures ,/,O varied for the different adsorbates: 10-11 ≤ ,/,O ≤ 0.98 for Ar and N2, 10-7 ≤ ,/,O ≤ 0.98 for CO2. As will be shown later the experimental data sets we used for validation of our approach were taken on a micro- and mesoporous sample and therefore mesopores were expected to provide a non-negligible amount of surface area. Since the monolayer formation of Ar and N2 on carbon takes place in a relative pressure region of 10-4 to 10-3,50 monolayer formation and micropore filling potentially occur in parallel. To take this correctly into account we included a set of slit-shaped mesopores from 20 to 80 Å (in steps of 2 Å) in the analysis of N2 and Ar adsorption isotherms.

2.4 Thermodynamic model for adsorption-induced deformation of disordered carbon structures To describe experimental strain data we invoke the adsorption stress model for adsorptioninduced deformation of disordered microporous carbons.7, 16 Within this model, it is assumed that the deformation of the microporous carbon matrix is sufficiently small to allow for the application of Hooke’s law for the volumetric strain ?v : 1 ∆ = − ∙ E (,/,O ) − , O

(16)

1 E (,/,O , d) @@ (d)dd O

(17)

?v (,/,O ) =

Here O is the specific volume of the microporous matrix, ∆ the change of the specific volume, the effective bulk modulus of the matrix and E the average adsorption stress within the microporous carbon matrix given by: E (,/,O ) =

Notably O represents the specific volume of the microporous matrix including pore and solid volume leading to an average pressure within the entire matrix not just the micropore volume. The adsorption stress E (d, ,/,O ) in a pore of width d at relative pressure ,/,O is derived from the grand potential:56 E (d, ,/,O ) = −

(d, ,/,O ) d ,|

(18)

where is the grand potential per unit area determined by the NLDFT model (Equation 1). From Equation 18 adsorption stress can be understood as the impulse of a pore to reduce its grand potential by deformation. Note that the change of the pore volume is fully attributed to the change of the pore width, while the surface area of the pore walls remains unchanged. An implicit assumption within the adsorption stress model is furthermore that adsorption-induced deformation is too small to influence the adsorption process; given that experimentally observed strains during gas adsorption are usually below 1 vol%, this assumption should be approximately meet. This is in line with theoretical investigations of the influence of adsorption-induced deformation on adsorption in micropores.6 Adsorption-induced stresses and strains in the plane of the pore walls and the corresponding effects on the adsorption process have been recently discussed in ref 57.


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3. Results and Discussion 3.1 Adsorption and stress isotherms: Examples for Ar (77 K) adsorption isotherms of different pore widths obtained by NLDFT calculations (Equation 12) are shown as average molar density Bv = 2wxyz{| /d of the adsorbate in Figure 1a, the corresponding adsorption stress isotherms (Equation 18) in Figure 1b. Respective plots for the adsorption of N2 (77 K) and CO2 (273 K) are given in the Supporting Information in Figures S1 and S2. The obtained adsorption isotherms are well in line with NLDFT calculations for the same adsorbate-temperature combinations performed earlier by Ravikovitch et al,50 who modeled the hard sphere repulsion of adsorbate molecules by the smoothed density approximation of Tarazona47 instead of the RLST version of the fundamental measure theory applied in this work (Equation 7). The calculated adsorption stress isotherms can be separated into the following groups: the pores exhibiting only positive adsorption stress (causing expansion), the pores exhibiting only negative adsorption stress (causing contraction, even though the stress isotherms are nonmonotonic), and the pores with non-monotonic adsorption stress isotherms with a transition from initial negative to positive adsorption stress with increasing relative pressure.

50 (a) Ar ρmol [mmol/cm³]

40 30 20 10 0 0.8

(b)

0.6 σa [GPa]

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0.4 0.2 0.0 -0.2 -0.4 -9

-8

-7

-6

-5

-4

-3

-2

-1

10 10 10 10 10 10 10 10 10

10

0

p/p0

Figure 1: (a) Adsorption (given as average molar density Bv of the adsorbate within the micropore) and (b) adsorption stress isotherms calculated by NLDFT for Ar adsorption at 77 K. The respective pore widths are (from left to right) 3, 4, 5, 6, 8, 10, 15 and 20 Å. As can be seen from Figure 1a, the filling of micropores smaller than 10 Å takes place in single step and the filling relative pressure ,@ /,O is defined from the inflection point of the step. Larger pores exhibit two or even three consecutive and distinct steps due to layering transitions 7 ACS Paragon Plus Environment

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preceding the pore filling; in the latter case, ,@ /,O corresponds to the last step on the isotherm. The relative filling pressure, the adsorption stress for relative gas pressures ranging from 10-9 (all pores empty) to 0.98 (all pores filled) and the average adsorbate density B = Bv ∙ % (% being the molar mass of the adsorbate, Table 1) at saturation pressure are shown for Ar adsorption (77 K) in Figure 2a, b and c, respectively, as a function of the internal pore width d. Respective plots for N2 (77 K) and CO2 (273 K) adsorption are given in the Supporting Information in Figures S3 and S4. Sharp increase of the filling pressure at the pore sizes around 11-12 Å (Figures 2a and S3a) is caused by the change of the filling mechanism with step-wise formation of the monolayer preceding the pore filling, as was already shown in earlier works.58 Noteworthy, the layer formation is an artifact of the assumption of smooth walls employed in the NLDFT model and significantly reduced for CO2 adsorption at 273 K (Figure S4a) due to the rather elevated temperature.

-1

10

(a) Ar

-3

pfilling/p0

10

-5

10

-7

10

-9

10

σa [GPa]

p/p0:

(b)

3

-9

10

-7

10 -5 10 -3 10 0.98

2 1 0

p/p0 = 0.98

(c) 1.8 ρmass [g/cm³]

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solid 1.6 1.4

liquid

1.2 2

4 w*

6

8

10 12 14 16 18 20

pore width w [Å]

Figure 2: (a) Pore filling pressure ,@ /,O , (b) adsorption stress E and (c) adsorbate mass density B calculated by NLDFT for Ar adsorption at 77 K as a function of the pore width d. d ∗ marks the minimum of ,@ /,O . In (c) the dashed horizontal lines indicate the densities of liquid Ar at 87 K59 and solid Ar at 40 K54.


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The adsorption stress for Ar (77 K) in carbon micropores close to saturation pressure (Figure 2b, ,/,O = 0.98) exhibits the well known damped oscillation with respect to pore width.40, 41 This oscillation is commonly interpreted as a result of the molecular packing of the adsorbate inside the pore:7, 42 if the pore width is commensurate with the molecular diameter of the adsorbate, the density of the adsorbate inside will be high and the short ranged hard sphere repulsion causes positive adsorption stress. On the contrary, inside a non-commensurate pore the adsorbed molecules are loosely packed and the long ranged attractive solid-fluid potentials dominate resulting in negative adsorption stress. This mechanism leads to a direct correlation of the adsorption stress and the average adsorbate density inside the micropore (Figure 2c), which also oscillates with respect to pore width due to the varying ratio of molecular size to pore width. The periodicity of both oscillations equals roughly the molecular diameter of the adsorbate.42 However, comparing adsorption stress and adsorbate density we find both oscillations shifted to each other by a difference in pore width, i.e., all maxima and minima of the adsorption stress take place at smaller pore widths than the respective maxima and minima of the density. This effect was also observed in earlier works.42 The adsorption stress and density oscillations become less pronounced with increasing pore width and temperature;42 the latter effect can be seen when comparing the calculated adsorption stress for N2 or Ar adsorption at 77 K (Figures 2b and S3b, respectively) to CO2 adsorption at 273 K (Figure S4b). Additionally the relative gas pressure has obviously a significant influence on the adsorption stress. For low relative pressure (,/,O = 10-9) all pores are essentially empty and there is no adsorption stress, while close to saturation pressure (,/,O = 0.98) all pores are filled and the oscillation is most pronounced. In between pores fill according to their filling pressure (Figure 2a) starting from pores of width d ∗(Ar) ≈ 3.35 Å, which divides the range of micropores in two groups: micropores larger than d ∗ exhibit a monotonic increase of ,@ /,O with pore width, but for pores smaller than d ∗ this trend is reversed (compare ref 58). Consequently for a given relative pressure ,/,O only pores with ,@ < , exhibit significant adsorption stress. Noteworthy, the smallest pores of the second group are usually neglected in adsorption studies, since it is commonly assumed that the adsorbate molecules do not penetrate into these pores; however, the solid-fluid Van-der-Walls interactions do not forbid filling of these pores, but rather make adsorption energetically less favorable as the pores become smaller. This is illustrated for Ar adsorption in Figure S5, where the external potential (Equation 10) is shown for pore widths comparable and smaller than d ∗ . As a consequence the filling of the smallest pores is shifted to elevated pressures. Quantitatively all extreme values of adsorption stress are obtained close to saturation at ,/,O = 0.98. We find the highest adsorption stress of approximately 3.8 GPa for Ar (77 K) adsorption at d = 2.5 Å; N2 (77 K) and CO2 (273 K) adsorption exhibit adsorption stresses of approximately 3.6 and 3.5 GPa, respectively, for a pore width of 2.7 Å. The minimum of Ar adsorption stress is -0.42 GPa for a pore width of 3.9 Å. Analogously, the N2 and CO2 adsorption pressure minimum is -0.38 GPa for pore widths of 4.3 and 4.2 Å, respectively. In parallel, the average Ar and N2 densities within the micropore oscillate between the values exceeding the free (bulk) solid and undercutting the free liquid; the density of adsorbed CO2 on the other hand always remains between respective values of free solid and free liquid. As to be expected, all adsorbate densities tend towards the density of the free liquid for increasing pore width. Comparing adsorption stress and relative filling pressures in Figures 2b and 2a, respectively, we find a remarkable correlation: the pore width d ∗ corresponds to a zero of the adsorption stress, which is independent of the relative gas pressure ,/,O . With respect to Equation 18 we therefore 9 ACS Paragon Plus Environment

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interpret d ∗ as the most comfortable pore,16 which is energetically ideal for the adsorbate and cannot further minimize its grand potential by a change of the pore width. This picture is well in line with the fact that pores of width d ∗ exhibit the lowest relative filling pressure. The numerical value of d ∗ depends on the adsorbate (d ∗ (Ar) ≈ 3.35 Å, d ∗ (CO2) ≈ 3.55 Å and d ∗ (N2) ≈ 3.6 Å) and corresponds approximately to the molecular diameter of the solid-fluid Van-der-Waals interaction E@ - again fitting the concept of a most comfortable pore size. The minima of the filling pressure of all adsorbates are shown together in Figure S6. Following this line of thought another important correlation is that all pores smaller than d ∗ exhibit exclusively positive adsorption stress, which increases drastically for proceeding adsorption. We interpret this finding in terms of adsorbate molecules squeezing into pores, which by common understanding would be too small for adsorption, i.e., in order to accommodate guest molecules these pores tend to expand and the adsorption stress in these pores is large and positive. A similar effect was described in ref 6 for flexible micropores. It appears only natural that such squeezing causes the relative filling pressure to increase rapidly as the pores become smaller (compare Figure S6).

3.2 Comparison with experimental data: The results derived by NLDFT calculations were tested against experimental data of adsorption and in-situ dilatometry measurements on a synthetic microporous carbon published in ref 31 . The experimental data for adsorption and corresponding linear strain from ref 31 are shown on logarithmic scale for CO2 in Figure 3, for N2 in Figure 4 and for Ar in Figure 5. Additionally, experimental N2 and Ar data are presented on linear relative pressure scale in Figure 6 and 7, respectively.


5 4

(a) CO2 experimental data adsorption PSD best fit PSD

3 2 1 0 0.6

∆L/L0 x 10³

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amount adsorbed [mmol/g]

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(b)

0.4 0.2 0.0 -0.2 -0.4 -5 10

-4

10

-3

10

-2

10

-1

10

p/p0

Figure 3: (a) Experimental CO2 adsorption isotherm and (b) corresponding strain isotherm from ref 31. Additionally shown are the predictions of the amount adsorbed by the integral adsorption equation (Equation 13) and the strain by the mechanical model (Equation 16) using the PSD obtained from analysis of the adsorption isotherms and the best fit PSD.


∆L/L0 x 10³

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8 (a) N2 6

experimental data prediction

4

prediction incl. mesopore surface area

2 0 2.5

0.10

2.0

0.05

(b)

0.00

1.5

-0.05

1.0

-10

-8

10

-6

10

10

0.5 0.0 -10

10

-8

10

-6

10

10

-4

10

-2

0

10

p/p0

Figure 4: (a) Experimental N2 adsorption isotherm and (b) corresponding strain isotherm from ref 31 (points). Additionally shown are the predictions of the amount adsorbed by the integral adsorption equation (Equation 13) and the strain by the mechanical model (Equation 16) using the best fit PSD (solid lines). The dotted line shows the prediction for the amount adsorbed if the best fit PSD is complemented by a 80 Å mesopore.


10 8

(a) Ar experimental data prediction

6 4

prediction incl. mesopore surface area

2 0 2.5

∆L/L0 x 10³

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0.1

(b)

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0.0

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1.0

10

-7

-5

10

0.5 0.0 10

-9

10

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10

-5

-3

10

10

-1

p/p0

Figure 5: (a) Experimental Ar adsorption isotherm and (b) corresponding strain isotherm from ref 31 (points). Additionally shown are the predictions of the amount adsorbed by the integral adsorption equation (Equation 13) and the strain by the mechanical model (Equation 16) using the best fit PSD (solid lines). The dotted line shows the prediction for the amount adsorbed if the best fit PSD is complemented by a 80 Å mesopore.


25

(a) N2

20 15

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experimental data prediction prediction incl. mesopore surface area

10 5 0 (b) 3

∆L/L0 x 10³

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2 1 0 0.0

0.2

0.4

0.6

0.8

1.0

p/p0

Figure 6: (a) Experimental N2 adsorption isotherm and (b) corresponding strain isotherm from ref 31 (points). Additionally shown are the predictions of the amount adsorbed by the integral adsorption equation (Equation 13) and the strain by the mechanical model (Equation 16) using the best fit PSD (solid lines). The dotted line shows the prediction for the amount adsorbed if the best fit PSD is complemented by a 80 Å mesopore.


25

(a) Ar

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experimental data prediction prediction incl. mesopore surface area

10 5 0 (b) 3

∆L/L0 x 10³

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2 1 0 0.0

0.2

0.4

0.6

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1.0

p/p0

Figure 7: (a) Experimental Ar adsorption isotherms and (b) corresponding strain isotherms from ref 31 (points). Additionally shown are the predictions of the amount adsorbed by the integral adsorption equation (Equation 13) and the strain by the mechanical model (Equation 16) using the best fit PSD (solid lines). The dotted line shows the prediction for the amount adsorbed if the best fit PSD is complemented by a 80 Å mesopore. An important aspect for the comparison of theory and experiment is the correlation of the volumetric strain of the microporous matrix ?v (Equation 16) and the linear strain ∆⁄O measured by the dilatometric technique on the monolithic sample. Here we assume the following: the sample investigated in ref 31 consists of interconnected microporous carbon particles, which form a 3dimensional mesoporous network. Both the particle network and the particles themselves are entirely isotropic and disordered with respect to their substructure. A schematic of the sample structure is shown in Figure S7. Based on this structure we assume a volumetrically isotropic deformation of the microporous particles, which in turn is transferred onto the particle network. In this case, ∆⁄O corresponds directly to the deformation of the microporous particles and ?v = 3∆⁄O , since ∆⁄O 10-4 expect for the capillary condensation in case of the N2 adsorption isotherm. Technically the mesopore surface area is included by adding a 80 Å slit-shape mesopore, which does not exhibit capillary condensation, to the best fit PSD and adapt its differential pore area. Next, we compare the experimental and the predicted strain isotherms for the adsorption of N2 and Ar at 77 K on the carbon sample. Here a purely computational issue arising from the finite number of calculated strain isotherms has to be taken into account; this is described in Figure S8 and the respective Figure caption. The predictions for the resulting strain of the sample are shown in Figures 4b, 5b, 6b and 7b along with the experimental data. Since the above discussed uncertainties of experimentally determined equilibrium pressures for ,/,O < 10-5 also apply to the comparison of experimental and theoretical predicted strain isotherms, we limit considerations to the reliable pressure range of ,/,O > 10-5 to begin with. In this range, the sample investigated expanded monotonically for increasing adsorption of Ar (Figures 5b and 7b) and N2 (Figure 4b and 6b). In both cases, the expansion is approximately proportional to the logarithm of the relative pressure. The adsorption stress model correctly predicts the monotonic expansion and its dependence on the relative pressure, however, for N2 adsorption the calculated expansion is quantitatively higher than in experiment, while for Ar adsorption the theoretical strain isotherm is lower than its experimental counterpart. Noteworthy the adsorption stress model predicts the expansion of the sample due to micropore filling to continue up to saturation pressure. For N2 adsorption the predicted expansion of 18 ACS Paragon Plus Environment

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the sample at saturation is about 10 % larger than its respective experimental value; for Ar adsorption it is 20 % lower (compare Figures 6b and 7b). This strongly favors the hypothesis that adsorption-induced deformation of the sample is dominated by the micropore filling over the whole relative pressure range, even when micropore filling is essentially completed and the changes of the amount adsorbed are insignificant. A rather fundamental issue, however, is the fact that the maximal strains for N2 and Ar adsorption at saturation are experimentally found to be nearly identical, while they differ significantly within the framework of the adsorption stress model despite identical structural and mechanical parameters. The reason for this is the difference of d ∗ for Ar, N2, and CO2, which approximately corresponds to the molecular Van-der-Waals diameters used in our calculations. While d ∗ (N2) and d ∗(CO2) are very similar, d ∗ (Ar) is significantly smaller. Since d ∗ marks the shift from pore widths exhibiting large positive adsorption stress and pore widths exhibiting oscillating adsorption stress (Figures 2b, S3b and S4b), Ar adsorption will cause less expansion at saturation than N2 or CO2 adsorption for all PSDs including pores on the scale of d ∗ . In parallel N2 and CO2 adsorption will result in very similar adsorption-induced deformations at saturation for a given PSD due to their similarity of d ∗ . Within the relative pressure range below 10-5, the sample investigated is experimentally found to contract but the minima of the experimental strain isotherms are rather small compared to the expansion observed upon the increase of adsorption. Qualitatively this behavior is correctly predicted by the adsorption stress model for the adsorption of N2 (see inset in Figure 4b). Also, the predicted extend of contraction is close to its experimental value. However, similarly to the comparison between experimental and theoretical N2 adsorption isotherms, the predicted relative equilibrium pressures are several orders of magnitude lower than the experimental ones. For Ar adsorption at ,/,O < 10-5 the agreement between experimental and theoretical strain is not as good as for N2. While the minimum of the strain isotherm is quantitatively correctly predicted, the adsorption stress model predicts a slight expansion of the sample preceding the contraction, which is not found in the respective experiment (compare inset in Figure 5b). The reason for the initial expansion is probably again the difference of d ∗ for CO2, N2, and Ar. The best fit PSD was determined from fitting of the experimental CO2 strain isotherm, and since CO2 and N2 exhibit very similar values for d ∗ , using this PSD one can expect a good prediction for experimental N2 data. On the other hand, for Ar adsorption, the best fit PSD leads to somewhat different results, which become most apparent when considering the slight contraction of the carbon matrix for low relative pressures.

4. Conclusions We employed NLDFT calculations to derive adsorption and adsorption stress isotherms for carbon micropores and the adsorbates N2 (77 K), Ar (77 K) and CO2 (273 K) including the micropores smaller than the nominal molecular of the adsorbates, which are typically disregarded. We found that the presence of these pores provide a clear correlation between pore filling pressure and adsorption stress. In particular there is a pore of width approximately equal to the Van-der-Waals diameter of the adsorbate, which exhibits the lowest relative filling pressure of all pores and is free of adsorption stress; therefore this particular pore width is the most comfortable pore size for the adsorbate. In contrast, smaller pores fill at higher relative pressure and exhibit exclusively large expansive adsorption stresses. We interpret this finding in terms of a forcing of the adsorbate molecules into the pore, which naturally requires higher gas pressures. Employing the NLDFT 19 ACS Paragon Plus Environment

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computations, we successfully modeled experimental data for CO2 adsorption and corresponding strain on a synthetic microporous carbon, which exhibited nonmonotonic deformation. Hereby, we derived a PSD for the sample investigated, which simultaneously matches the adsorption and the strain isotherm. Applying the model parameters obtained from experimental CO2 data, we predicted adsorption and adsorption induced strain for N2 and Ar. Comparing these predictions with the respective experimental data, semi-quantitative agreement was found within reasonable boundaries for relative pressures above 10-5. This allows for the conclusion that adsorption stress in micropores is the dominant mechanism for the strain of the sample even in the relative pressure regime above 0.1, where the micropore filling is essentially completed and the adsorption isotherm reaches a plateau. For relative pressure ,/,O < 10-5, quantitative comparison between experiment and theory reveals significant deviations. In case of N2 adsorption, the deviations between experimental and calculated strain can be attributed to the experimental difficulties to obtain reliable equilibrium gas pressures in the low relative pressure regime. For Ar adsorption, we ascribe the discrepancies between experimental and theoretical strain isotherms to the difference of the NLDFT Van-der-Waals diameter between Ar on the one hand and N2 and CO2 on the other hand.

Associated Content Supporting Information NLDFT results for N2 adsorption at 77 K and CO2 adsorption at 273 K. External potentials in micropores for Ar adsorption for pore widths comparable and smaller than the nominal molecular diameter. Comparison of the filling pressure dependence of all adsorbates on the pore width. Scheme of the structure of the carbon sample. Smoothing of the average adsorption stress E in the interval 2 Å < d < d ∗ . This material is available free of charge via the Internet at http://pubs.acs.org.

Author Information Corresponding Authors *E-mail: [email protected] (C.B.) *E-mail: [email protected] (AVN) *E-mail: [email protected] (G.R.)

Acknowledgements The authors are grateful to Yangzheng Lin for help with NLDFT calculations. Christian Balzer acknowledges support by the Deutscher Akademischer Austauschdienst (DAAD – German Academic Exchange Service). A.V.N. acknowledges support from the Rutgers NSF ERC on Structured Organic Particulate Systems.


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