Capillary Condensation and Evaporation in Irregular Channels

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Capillary Condensation and Evaporation in Irregular Channels: Sorption Isotherm for Serially Connected Pore Model Daniel Schneider, and Rustem Valiullin J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b03626 • Publication Date (Web): 07 Jun 2019 Downloaded from http://pubs.acs.org on June 8, 2019

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The Journal of Physical Chemistry

Capillary Condensation and Evaporation in Irregular Channels: Sorption Isotherm for Serially Connected Pore Model Daniel Schneider and Rustem Valiullin∗ Felix Bloch Institute for Solid State Physics, University of Leipzig, Linn´estr. 5, 04103 Leipzig, Germany E-mail: [email protected]

Abstract Geometrical disorder can strongly impact phase equilibria of fluids in mesoporous solids. There is insufficient knowledge of how the structural disorder results in the emergence of the cooperativity effects in phase transitions. To tackle this problem, understanding of the complex interplay between nucleation and phase growth and the pore space morphology is needed. We use statistically disordered chains of serially connected single pores with varying pore sizes to mimic geometric disorder and solve the problem using a statistical thermodynamics approach. As the main result, we derive the exact solution for the average phase composition at any thermodynamic condition including all states within the hysteresis region, i.e. the entire family of the sorption isotherms including the scanning isotherms. We show that our approach correctly reproduces the results of computational modeling using the mean field theory of lattice gas in irregular model pores. The theory developed is directly applicable for the analysis of phase equilibria in materials with tubular pores, such as MCM-41 and

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SBA-15, but can also be used to gain deeper insight into phase behavior in mesoporous solids with random pore networks.

Introduction Understanding phase equilibria of fluids confined in small pores is of importance in many areas of science and technology, including chemical engineering, environmental and applied sciences, medicine, etc. Some important examples are freezing of water in porous dust particles as an inhibitor of ice formation in clouds, 1 friction induced by capillary condensation, 2–4 sound and light propagation in partially-saturated porous solids, 5,6 optical switching by capillary condensation, 7 altering of heat transfer at nanoscale, 8 and material deformation due to capillary condensation. 9 Many aspects of these phenomena occurring in pore spaces with simple pore morphologies, such as in cylindrical channels or in slit-like pores (referred to in what follows as single pores), have extensively been addressed in the literature (see, for example, recent reviews 10–15 ). Summarizing the most essential points of these studies, three important aspects may be discerned. Foremost, the phase coexistence lines for material in nanoscale pores are found to be shifted with respect to that of bulk fluids. These shifts scale typically in proportion with the inverse pore size. The second common observation is that phase transitions of confined materials typically exhibit irreversibility. And finally, the boundary conditions at the pore openings decide on the transition mechanism by either introducing or by removing the nucleation barriers. In materials with complex pore morphologies, phase equilibria become increasingly intricate. 16–18 First of all, the emerging distribution of the confinement sizes leads to a respective distribution of the transition points, such as the condensation pressures. For a collection of separated single pores with a distribution of the pore sizes, the respective spread of the transition points can easily be quantified. This becomes possible due to the fact that the boundary conditions at the pore openings of each pore in the ensemble are identical and thus

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the overall transition behavior becomes simply cumulative of the behaviors in each single pore. Any interconnection between the single pores complicates the problem notably. In this case, the transition mechanism in a selected pore becomes, in addition, determined by the phase state in the adjacent pore. Convincing evidences for this scenario have been provided by the studies of the so-called ink-bottle pore systems. 19–25 In materials with geometric disorder, this coupling between phase states in different parts of the pore network gives rise to the strong cooperative character of the phase transitions and, hence, to very complex phase equilibria. Theoretical description of phase transitions under these conditions is a challenging problem. 26–34 In the recent decades, there was a growing evidence that, in materials with seemingly ideal single pore structures, the phase transitions may exhibit some features typical for disordered materials. 35–43 In particular, the sorption isotherms in MCM-41 and SBA-15, the materials possessing channel-like pores, often reveal asymmetry between the transition branches and, most importantly, the scanning behavior is found to be untypical of single pore materials. 35,38,44 Both these features, typically observed in mesoporous solids with complex pore morphologies, such as Vycor porous glass, so far were attributed to network effects. 17,33,45,46 Establishing the underlying reasons for their observations in materials with tubular pore structure is, thus, essential for both validating and improving theoretical models for phase equilibria on a single pore level and for better understanding the real structure of materials under study. One of the hypothesis suggested is that corrugations of the pore walls and undulations along the channel axes can be responsible for the phenomena observed. 41–43,47–49 Computer-based numerical analyses using linear pores composed of cylindrical pore sections with varying diameters indeed confirmed the emergence of the patterns typical of real, geometrically disordered porous solids. 50–52 It was shown recently that, for statistically disordered linear chains of pores with varying pore sizes, in what follows referred to as serially connected pore model (SCPM), the phase equilibria can be obtained analytically using a statistical thermodynamics approach. 34 In

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particular, an approximate solution for the boundary phase transitions and the scanning transitions were obtained. The sorption isotherms derived agreed qualitatively well with the experimental observations. However, the approximation made had two essential drawbacks. First, the approximate isotherm did not allow for an accurate quantitative analysis, especially for short chains of single pores, which is the case in MCM-41 or SBA-15 materials. It is worth noting that having an accurate solution is important not only from the material characterization perspective, but also the accurate model for one-dimensional disordered chains provides a basis for quantitative analysis of network effects in real materials. Secondly, and most importantly, the earlier theory did not provide how different phases, capillarycondensed and gaseous, are distributed along the pore space. The latter, as now provided in the present work, opens new avenues for addressing complex physical phenomena depending on the phase distribution properties. As the most relevant examples, light scattering and mass transfer in partially-filled porous solids, which are still far from being quantitatively described, may be mentioned. In the present work, we provide the details how the sorption isotherms can be obtained for the SCPM and prove the accuracy of the solution obtained by numerical solution of the mean field theory for lattice gas in the model pore systems. Notably, the theory outlined here may easily be adopted for describing freezing and melting transitions in porous materials, mercury intrusion and extrusion and also related structural transitions exhibiting pore-size dependent properties. The derivation details may also be important for those aiming at analysis of scattering properties under conditions of phase coexistence in mesoporous solids.

Results Pore model and transition mechanisms The pore spaces in our work are represented by chains of joined cylindrical pore sections with different diameters x and equal length l (see Fig. 1), in what follows referred to as 4


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Figure 1: Schematic illustration of a discrete model to describe statistically disordered onedimensional pores. serially connected pore model (SCPM). The section volumes are distributed according to a normalized distribution function φ(x), the so-called pore size distribution (PSD). It is assumed that the diameters x of two adjacent channel sections are statistically independent. The total length of a channel, L, is measured in units of l and depicts, thus, the number of statistically uncorrelated pore sections. During sorption in mesopores, vapor (grey) and condensed (blue) phase may coexist, forming gaseous and liquid domains, i.e. continuous domains containing either capillary-condensed liquid or gaseous phase coexisting with a liquid film on the channel walls. The length of such phase domains in units of l is denoted by λ. Large ensembles of such channels with different realisations of disorder are considered allowing for a statistical average. With variation of the gas pressure, the phase equilibria in the pores depend on the governing phase transition mechanisms. It is considered that the phase composition may change by nucleation and by phase growth as explained in more details in what follows. As for the first order phase transitions, the lack of nuclei of the new phase may prevent the phase transition, trapping the old phase in a metastable state. In this case, a nucleation event is needed for the transition to occur. For capillary condensation in pores, this mechanism corresponds to the phenomenon called liquid bridging, while for evaporation it refers to cavitation. We assume that, as soon as a nucleus is formed in a pore section, phase transition in the entire pore section is stimulated. In the following, this mechanism is referred to by nucleation and denoted with the subscript n. Phase transformation in some pore sections 5



may also be triggered by an already existing nucleus. The role of a nucleus can be played by liquid or gas phase already formed in a pore section directly adjacent to the pore section in which phase transformation is considered. Alternatively, it may be the new phase supplied at the channel openings. Because metastability in this case is removed, these transitions occur in thermodynamic equilibrium. The respective phenomena in sorption are associated with advanced condensation or evaporation via gas invasion. In the remainder of this work, this mechanism is termed growth and is indicated with the subscript g. Whether the capillary transition in an arbitrarily selected pore section occurs by nucleation or growth is determined by thermodynamic conditions, such as temperature and gas pressure, the section diameter x, and the boundary conditions, namely, the phase state in the adjacent pore sections. The latter introduces a nearest-neighbour coupling in the chain and leads to the cooperativity effects in the phase transitions. The phase behavior in separate, single pore sections is described by the so-called kernels, i.e., theoretically or experimentally obtained sets of sorption isotherms for varying pressure p and different pore diameters x. In the following, the two nucleation kernels are denoted with θn (x, p) for liquid bridging and θn0 (x, p) for cavitation. Because of reversibility in thermodynamic equilibrium, the growth kernels for advanced condensation and gas invasion coincide, i.e. θg (x, p) = θg0 (x, p). From these kernels, also the critical diameters for each transition mechanism can be obtained, dividing the pore sections in terms of their pore sizes into two parts, namely, in which nucleation and growth may occur and in which not. These critical diameters are denoted with xn (p), xg (p) = x0g (p), and x0n (p) for liquid bridging, advanced condensation/gas invasion, and cavitation, respectively. For readability, in what follows the condensation process and the pore sections containing the capillary condensed phase are termed with filling/filled, while evaporation and the pore sections containing the gaseous phase coexisting with the physisorbed film on the channel walls denoted with emptying/empty. Also, the quantities corresponding to emptying/empty are labeled with the prime symbol (0).

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SCPM adsorption isotherm Based on the independent domain model (IDM), the general adsorption isotherm (GAI) equation is widely accepted to relate sorption in a disordered porous solid to the kernels of the independent pore sections it is comprised of. 53 For the adsorption boundary curve it can be expressed as ads θGAI (p)

Z =

θn (x, p)φ(x)dx.

(1)

The only transition mechanism considered by Eq. (1) in cylindrical pores, as often assumed in the literature, is nucleation. Similarly, if a spherical pore approximation is used, metastability is inherently removed and only the equilibrium transition condition is considered. In this case, θn (x, p) is replaced by θg (x, p). Since IDM treats the phase behaviors in all pore sections independently, stimulated phase growth along the chain becomes ineffective. Hence, Eq. 1 underestimates the amount adsorbed θ in the pore networks. The same scenarios apply also to the desorption transition. SCPM introduces the pore interconnectivity and, thus, takes phase growth into account with the help of a correction term to Eq. 1. In particular, one may write the SCPM isotherm for adsorption as θ

ads

(p) =

(ads) θGAI

Z +

(θg − θn )ψdx,

(2)

where ψ indicates PSD of the filled sections only. Eq. 2 can be modified to

θ

ads

Z (p) =

θg ψdx + θn ψ 0 dx,

(3)

where ψ 0 (x, p) denotes PSD of the empty sections, and ψ + ψ 0 = φ. For desorption, the SCPM isotherm can be expressed in a similar way as

θ

des

Z (p) =

θn0 ψdx + θg0 ψ 0 dx.

(4)

Physically, Eqs. (3) and (4) divide the integral over the pore sections into two parts. The 7



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first represents the contribution of all filled pore sections, while the second one gives the contribution of the empty sections amounting in just adsorbed layers on the pore walls. In what follows, we first describe how to obtain PSDs ψ and ψ 0 for the main adsorption and desorption transitions. Note that, as soon as ψ and ψ 0 are established, the phase equilibria become fully described.

Adsorption boundary curve For obtaining the adsorption isotherm, one starts from the initial state with all pores being in the empty state. During a quasistatic rise of gas pressure, the pore sections fill gradually. Recalling that the critical pore diameters for nucleation (liquid bridging) and for growth (advanced adsorption) are xn (p) and xg (p), respectively, and xg (p) > xn (p) ∀ p, PSD of the filled sections can be found as

ψ(x, p) = φ ×

    1     Ptr       0

x ≤ xn xn < x ≤ xg .

(5)

xg < x

PSD of the empty pore sections can simply be obtained by ψ 0 = φ − ψ. Eq. (5) implies that filling of every pore section with a diameter smaller than xn is triggered by nucleation. For xn < x ≤ xg only a certain fraction Ptr (p) of the pore sections are filled by growth from the adjacent sections already filled with the condensed phase. Hence, Ptr (p) represents the probability that a randomly selected pore section is connected directly or indirectly, via other pore sections with x ≤ xg , to a section with the condensed liquid. Finally, all pore sections with x > xg remain empty because they cannot be filled by either mechanism. In Eq. (5), Ptr is the sole unknown quantity. To obtain Ptr , we start with introducing the mean probabilities Pn and Pg that an arbitrarily selected pore section has sufficiently small size to allow for nucleation or growth, respectively. For cylindrical geometry of the

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pore sections, both probabilities can be obtained as R Pn (p) =

x−2 φdx x≤xn R x−2 φdx

(6)

and R Pg (p) =

x≤xg

R

x−2 φdx

x−2 φdx

.

(7)

Note that these probabilities turn out to be the cumulative probabilities of normalized number distribution functions. The mean total probability that a section is filled by either of the mechanisms is R −2 x ψdx = Pn + Ptr (Pg − Pn ). P (p) = R −2 x φdx

(8)

Alternatively, P can be expressed as

P (p) = Pg Pst .

(9)

Eq. (9) implies that two conditions need to be fulfilled for a pore section to fill: (i) The section has to be narrow enough for capillary condensation to be possible, i.e. x ≤ xg . This is simply Pg . (ii) The section may be either small enough for a nucleation event to occur (x ≤ xn ) or, otherwise, needs to have access to at least one nucleus at its boundaries. The respective, combined probability is denoted by Pst (‘stimulated’). In order to obtain Pst , the length distribution fg (λ) of the continuous domains composed of the pore sections meeting the condition (i) needs to be found. Here, λ denotes the length of the domains in units of number of the pore sections. Combinatorial analysis yields

fg (λ, p) =

   Pgλ (1 − Pg ) [2 + (L − λ − 1)(1 − Pg )] λ < L

Pg2

1 + LPg − LPg2   PgL

λ=L

9


.

(10)


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Any such domain fulfilling the condition (i) can only be filled if at least one nucleation event can occur within the domain, i.e. it has to entail at least one pore section with x < xn . This nucleus will stimulate filling of the entire domain by growth. The corresponding probability Pst (λ, p) for a domain of length λ to be filled is Pst (λ, p) = 1 − fbin

Pn 0, λ, Pg

Pn =1− 1− Pg

λ ,

(11)

where fbin denotes the binominal distribution. With the help of Eqs. (10) and (11), the mean probability Pst is found as an average

Pst (p) = hPst (λ)iλfg (λ) ,

(12)

where the mean value operator is defined as P

X(y)F (y) P . y F (y)

y

hX(y)iF (y) =

(13)

By comparing Eqs. (8) and (9) one finds

Ptr (p) =

Pst Pg − Pn . Pg − Pn

(14)

With Ptr being obtained, PSDs of the filled and empty sections become fully determined. Ultimately, the adsorption boundary curve can be obtained with Eq. (3). It is worth noting that, in the limit of L = 1, the SCMP isotherm (Eq. (3)) naturally converges to that of IDM as given by Eq. 1. See for the proof the appendix. To complement the description of the phase state in the pore, the length distribution f (λ, p) of the filled domains can be obtained via Pst (λ)fg (λ) f (λ, p) = PL . λ=1 Pst (λ)fg (λ)

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(15)

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Along with the average phase composition, as expressed by the adsorption isotherm, Eq. 15 provides the complete statistical description of the phase equilibria along the adsorption boundary curve.

Desorption boundary curve The analysis for the desorption branch is performed in the same way. For this, recall that the critical pore diameters for nucleation (cavitation) and growth (gas invasion) are, x0n (p) and x0g (p), respectively. Initially, all pore sections are filled with the capillary condensed liquid. Upon a quasistatic decrease of the gas pressure, the pore sections empty gradually. PSD associated with the empty sections only is

ψ 0 (x, p) = φ ×

    0    

x ≤ x0g 0

Ptr       1

x0g < x ≤ x0n ,

(16)

x0n < x

where x0n (p) > x0g (p) ∀ p, and PSD of the filled sections is found as ψ = φ − ψ 0 . Every pore section larger than x0n is emptied by cavitation of gas bubbles. For the sections in the range x0g < x ≤ x0n only the fraction Ptr0 can emptied by gas invasion, namely, only those sections in contact with the pore sections in the empty state. All pore sections with x ≤ x0g remain filled. Similar to adsorption, an expression for Ptr0 is the unknown to be obtained. Similar to Eqs. 7 and 6, we introduce the mean probabilities Pn0 and Pg0 as R Pn0 (p) =

x−2 φdx x>x0n R x−2 φdx

(17)

and R Pg0 (p)

=

x>x0g

R

11

x−2 φdx

x−2 φdx

.


(18)


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Now, Ptr0 , PSDs of the empty and filled sections, and f 0 (λ, p) can be obtained with essentially the same procedure as for the adsorption boundary curve by replacing all quantities corresponding to the filled phase with those of the empty phase (X → X 0 ). The only adjustment needs to be made is related to the fact that, at the channels openings, there is direct contact to the gas phase at all pressures. This provides another pathway for emptying in addition to that triggered by cavitation. With this, the total probability Pst0 (λ) for a domain of length λ to be emptied is

0 0 0 0 0 0 Pst0 (λ, p) = Pst, n ∨ Pst, g = Pst, n + Pst, g − Pst, n Pst, g .

(19)

0 0 The probability Pst, n for cavitation in the domain and the probability Pst, g to have access

to the external vapor phase are

0 Pst, n (λ, p)

P0 = 1 − 1 − n0 Pg

λ (20)

and

0 Pst, g (λ, p) =

2 , 2 + (L − λ − 1)(1 − Pg0 )

(21)

respectively. Ptr0 , the quantity completely determining the solution of the problem, is given by

Ptr0 (p)

Pst0 Pg0 − Pn0 = Pg0 − Pn0

(22)

with

Pst0 (p) = hPst0 (λ, p)iλfg0 (λ) ,

(23)

where fg0 (λ) is given by Eq. 10 with Pg being replaced by Pg0 . Finally, the desorption

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boundary curve can be obtained by inserting PSDs for the empty and filled sections into Eq. (4).

Desorption scanning curves Let us now consider the states obtained within the hysteresis loop and which are achieved, for example, by performing desorption upon incomplete adsorption. The desorption scanning curves can be obtained with essentially same consideration as the boundary isotherms, but with the correspondingly adjusted initial conditions. Let us denote with p0 the pressure attained along the boundary adsorption branch, where the desorption scan is initiated. At this point, a complex distribution of the empty and filled domains along the channels is found. The corresponding PSDs of the filled and empty pore sections at p0 we denote with ψ0 (x) (Eq. 5) and ψ00 (x) = φ−ψ0 , respectively. The lengths Λ of the filled domains subject to emptying during the desorption scan, Λ ∈ N≤L , are distributed according to f0 (Λ), as given by Eq. (15). From this initial phase state, the desorption scan is performed with decreasing pressure. At a pressure p < p0 , PSDs of the filled and empty sections along the desorption scanning curve can be obtained as

ψ(x, p, p0 ) = ψ0 − ∆ψ 0

(24)

ψ 0 (x, p, p0 ) = ψ00 + ∆ψ 0 ,

(25)

where

∆ψ 0 (x, p, p0 ) = ψ0 ×

    0    

x ≤ x0g 0

Ptr       1

x0g < x ≤ x0n .

(26)

x0n < x

In the equations above, only the mean fraction of the pore sections emptied by growth, 13



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Ptr0 is unknown. In order to derive Ptr0 , the mean probabilities for an arbitrary pore section to empty by nucleation and growth are needed. In spirit of Eqs. 17 and 18, they are found as x−2 ψ0 dx x>x0n R x−2 ψ0 dx R x−2 ψ0 dx x>x0g R

Pn0 (p, p0 ) = Pg0 (p, p0 ) =

R

x−2 ψ0 dx

.

(27) (28)

The mean total probability that a section is emptied by either one of the mechanisms is found as R −2 x ∆ψ 0 dx R P (p, p0 ) = = Pn0 + Ptr0 (Pg0 − Pn0 ), −2 x ψ0 dx 0

(29)

where

Ptr0 (p, p0 ) =

Pst0 Pg0 − Pn0 Pg0 − Pn0

(30)

(see, for the derivation, the discussion preceding Eq. 14). In order to obtain Pst0 in Eq. 30, the length distribution of the continuous domains which can be formed by gas invasion needs to be found. For this purpose, desorption in each of the initially filled domains drawn from f0 (Λ) need to be treated individually. In such a domain of length Λ, the lengths λ ≤ Λ of the smaller domains that may empty during the desorption scan, are distributed as

fg0 (λ, Λ, p, p0 ) =

   Pg0λ (1 − Pg0 ) 2 + (Λ − λ − 1)(1 − Pg0 ) λ < Λ

1 × Pg02 + ΛPg0 − ΛPg02   Pg0Λ

.

λ=Λ (31)

For these domains to empty, at least one nucleation event within the domain has to occur. 0 The probability of this event is Pst, n ). Alternatively, it can empty if contact to the empty 0 phase is provided at the boundary. The respecttive probability is Pst, g . The combined

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probability is then

0 0 0 0 0 0 Pst0 (λ, Λ, p, p0 ) = Pst, n ∨ Pst, g = Pst, n + Pst, g − Pst, n Pst, g .

(32)

With respect to Eq. (32), the probability for at least one nucleation event to occur is calculated as 0 Pst, n (λ, p, p0 )

λ Pn0 =1− 1− 0 . Pg

(33)

The probability for contact with an adjacent empty phase can be obtained with

0 Pst, g (λ, Λ, p, p0 ) =

2 . 2 + (Λ − λ − 1)(1 − Pg0 )

(34)

Finally, the mean probability Pst0 (p, p0 ) is found as Pst0 (p, p0 ) = hhPst0 (λ, Λ)iλfg0 (λ,Λ) iΛf0 (Λ) .

(35)

By inserting Eqs. (35) and (30) into Eq. (26) PSDs of the filled and empty pore sections become fully determined. The desorption scanning curve can now be obtained using Eq. (4). Additionally, the length distribution of the empty domains formed during the scan from p0 to p can be obtained with * f 0 (λ, p, p0 ) =

Pst0 (λ, Λ)fg0 (λ, Λ)

+ (36)

PL

0 0 λ=1 Pst (λ, Λ)fg (λ, Λ)

Λf0 (Λ)

Adsorption scanning curves A similar strategy is used for the derivation of the adsorption scanning curves. Consider an adsorption scan starting at pressure p0 on the desorption boundary curve. Let us denote PSDs of the empty and filled pore sections at this initial point with ψ00 (x), as given by Eq. (16), and ψ0 (x) = φ − ψ00 , respectively. The lengths Λ ∈ N≤L of the empty domains,

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subject to filling during the subsequent pressure increase, are distributed according to f00 (Λ). This distribution function is given by Eq. (15), but with correspondingly replaced parameters relevant for desorption. At a pressure p > p0 , PSDs of the filled and empty pore sections can be obtained as

ψ(x, p, p0 ) = ψ0 + ∆ψ

(37)

ψ 0 (x, p, p0 ) = ψ00 − ∆ψ

(38)

where

∆ψ(x, p, p0 ) = ψ00 ×

    1     Ptr       0

x ≤ xn xn < x ≤ xg .

(39)

xg < x

Once again, to calculate Ptr , we introduce the mean probabilities for an arbitrary pore section to fill by nucleation and growth as x−2 ψ00 dx x≤xn R = x−2 ψ00 dx R x−2 ψ00 dx x≤xg R

Pn (p, p0 )

Pg (p, p0 ) =

R

x−2 ψ00 dx

,

(40) (41)

respectively. In line with the consideration presented in the preceding sections, Ptr (p, p0 ) is found as

Ptr (p, p0 ) =

Pst Pg − Pn . Pg − Pn

(42)

In order to obtain Pst , the length distribution of the continuous domains found in the empty state and fulfilling the condition x ≤ xg needs to be found. For this purpose, adsorption in each of the initially empty domains drawn from f00 (Λ) may be treated individually.

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In such a domain of length Λ, the lengths λ ≤ Λ of the domains we are looking for are distributed as

fg (λ, Λ, p, p0 ) =

   Pgλ (1 − Pg ) [2 + (Λ − λ − 1)(1 − Pg )] λ < Λ

Pg2

1 × + ΛPg − ΛPg2   PgΛ

. (43)

λ=Λ

In these domains, at least one nucleus is needed to fill the entire domain with the condensed liquid. There are two scenarios how this can be accomplished: Either a liquid bridge can be formed within a domain (the respective probability is Pst, n ) or direct contact to a filled phase domain at the boundary is given (the respective probability is Pst, g ). The combined probability is then

Pst (λ, Λ, p, p0 ) = Pst, n ∨ Pst, g = Pst, n + Pst, g − Pst, n Pst, g .

(44)

The probability for at least one nucleation event to occur is λ Pn Pst, n (λ, p, p0 ) = 1 − 1 − . Pg

(45)

The probability for having contact with an adjacent filled section is

Pst, g (λ, Λ, p, p0 ) =

2b0 , 2 + (Λ − λ − 1)(1 − Pg )

(46)

where b0 accounts for the existence of a phase interface at the boundary of the domains to be filled as follows. In the interior of a channel with the alternating filled and empty domains, each empty domain is in contact with the filled domains on both sides. However, an empty domain at either of the channel ends is in contact with only one filled domain due the external bulk phase on the other side. Thus, with n00 indicating the mean number of the empty domains in a channel at p0 , we define b0 as b0 = (n00 − 1)/n00 . n00 itself can be found by the relation between the combined length of the empty domains and the mean length of 17



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one empty domain, i.e. n00 = P00 L/hΛi. Hence, b0 can be expressed as R L 1 x−2 φdx X 0 hΛi R Λf (Λ)dΛ. b0 = 1 − 0 = 1 − P0 L L x−2 φ00 dx Λ=1 0

(47)

Finally, the mean probability Pst (p, p0 ) is obtained as a double average

Pst (p, p0 ) = hhPst (λ, Λ)iλfg (λ,Λ) iΛf00 (Λ) .

(48)

Inserting Eqs. (48) and (42) into Eq. (39) fully determines PSDs of the filled and empty sections, and the adsorption scanning curve is obtained using Eq. (3). Additionally, the length distribution of the filled domains formed during the scan can be obtained with * f (λ, p, p0 ) =

Pst (λ, Λ)fg (λ, Λ) PL

λ=1

Pst (λ, Λ)fg (λ, Λ)

+ .

(49)

Λf00 (Λ)

Discussion To verify SCPM, we have used mean field theory of lattice gas model to explore the phase equilibria in long disordered pores generated using computer-based algorithms. In particular, long channels consisting of L = 500 interconnected cylindrical pore sections with their diameters drawn randomly from a gamma distribution with the mean µ = 12 nm, shape factor k = 6, and two cut-offs taken at 3 nm and 28 nm were generated. Note that the diameters drawn were discretized and varied in steps of 1 nm. Each of the pore sections had a length of 25 nm, which gives a total channel length of 12.5µm. To obtain the sorption isotherms, the mean field theory was used. For a more accurate statistical representation the results were averaged over different channels with different disorder realizations. Because of extremely long computations, only an average over 8 sample pores was managed. The simple cubic lattice gas model in an external filed with nearest-neighbour interactions was employed to describe the phase behavior in the thus obtained channels following the work

18


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by Woo et al. 54 The configurational energy was calculated as

H(ni ) = −

X εX ni nj + ni φi , 2 i

(50)

hiji

where ni represents the occupation number on a lattice coordinate vector i, hiji denotes the sum over all nearest-neighbour pairs, ε is the intermolecular interaction constant, and φi models interaction of molecules with the surface. The grand canonical potential in the mean field approximation is

Ω=−

X X 1 X εX ρi + ρi φi − µ [ρi ln ρi + (1 − ρi ) ln(1 − ρi )] , ρi ρj + 2 kT i i i

(51)

hiji

where ρ = hni i is the ensemble average, and µ is the chemical potential related to the gas pressure. By minimizing Ω, the equilibrium densities at each T , V and µ are found as ( ρi =

1 + exp

X 1 [−ε ρj + φi − µ] kT j

)!−1 .

(52)

Sorption kernels First, the mean field theory was applied to obtain the respective kernels for the phase transition mechanisms. For that purpose, ideal cylindrical channels of different diameters were used. Fig. 2 a exemplifies the kernel isotherms for all phase transition mechanisms obtained for a pore of 10 nm diameter at T /Tc = 3/4. To obtain the kernel isotherms for liquid bridging, θn (x, p), open-ended pores in contact to a bulk gas phase on both sides were used. Since this morphology lacks a nucleus, during adsorption capillary condensation can only originate from the nucleation of a liquid bridge. The advanced adsorption and desorption kernels, θg (x, p) and θg0 (x, p), were acquired with the capped pore geometry closed by a pore wall on one side and open towards a bulk gas phase on the other. Here, the transitions

19



Figure 2: a: Sorption isotherms exemplifying liquid bridging (solid black line), advanced adsorption and gas invasion (dashed red line), and cavitation (dotted blue line) as obtained with mean field theory for an ideal cylindrical pore with a diameter of 10 nm. The other figures show the kernels for liquid bridging (θn (x, p), b), advanced adsorption and desorption (θg (x, p) = θg0 (x, p) , c), and cavitation (θn0 (x, p), d). In b - d, the different curves represent different pore diameters x, starting with 3 nm in the upper left corner and increasing to 28 nm towards the bottom right corner. All curves were obtained at T /Tc = 3/4. occur via growth from a nucleus either the capped end for condensation or the open end for evaporation. Condensation and evaporation via growth are reversible, i.e. θg = θg0 . Here, the capped end pore was used as an approximation of a situation where an adjacent pore section contains the capillary condensed liquid. The cavitation kernel, θn0 (x, p), was obtained upon desorption with a closed pore geometry, making intrusion of the gas phase impossible. Thus, here the only mechanism to evaporate is the nucleation of a gas bubble. Figs. 2 b, c, and d show all kernels obtained. Note that surface layer adsorption is observed in the 20


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Figure 3: Critical pore diameters for the different phase transition mechanisms upon variation of the gas pressure at T /Tc = 3/4. The black dots indicate the critical diameters for liquid bridging (xn ), the red triangles for advanced adsorption and gas invasion (xg = x0g ), and the blue diamonds for cavitation (x0n ). lower pressure regime. For all isotherms shown two distinct states with a sharp transition in between can be seen, the gaseous state with adsorbed surface layers (empty) and the capillary-condensed state (filled) for pressures close to the saturated vapor pressure. Fig. 3 shows the critical pore diameters for all phase transition mechanisms, namely, xn for liquid bridging, xg = x0g for advanced adsorption and gas invasion, and x0n for cavitation.

Transitions in disordered channels Figure 4 shows the MFT results obtained with the disordered channels, in which the pore section diameters are distributed according to the gamma distribution as described earlier. A relatively broad hysteresis loop is formed primarily due to strong pore blocking. This becomes evident by comparing the MFT results to the prediction of the IDM theory shown by the lines. The latter predicts significantly narrower hysteresis between adsorption and desorption and fails to reproduce both transition branches. 21



Figure 4: Adsorption and desorption boundary curves obtained from the MFT calculation for the disordered pores (symbols) and as predicted by the independent domain model (solid lines). In contrast to IDM, as demonstrated by Figures 5a, SCPM shows almost perfect match between theory and the MFT results (note that the step-like behavior seen in the theoretical predictions, results from a discrete set of the kernels and the section diameters used). Moreover, the SCPM theory reproduces perfectly not only the boundary transitions, but also the scanning behavior. This is exemplified in Figure 5 showing also the desorption scans obtained using MFT and SCPM. Additional proof of the validity of SCPM is demonstrated in Figure 6 evidencing that SCPM is capable of reproducing any state within the hysteresis loop obtained, for example, for more complex partial isotherms, such as scanning loops.

Conclusions In the present work we present statistical thermodynamics theory for phase equilibria in irregular one-dimensional pores. Without any rigorous restrictions it can be applied to describe gas sorption in porous solids with unidirectional pores, such as MCM-41 or SBA-15. By directly incorporating the disorder effects, modeled by variation of the channel diameter along the channel axes, and by allowing for the interplay between nucleation and phase growth, the SCPM theory is found to reproduce the majority of the experimental findings reported 22


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Figure 5: a: Comparison of the sorption boundary curves obtained using MFT (symbols) for the disordered pores and the prediction of SCPM (lines), Eqs. (3) and (4). For SCPM, the kernels shown in Figure 2 were used. b and c: Desorption and adsorption scanning curves obtained using MFT for the disordered pores and the predictions of SCPM (solid lines), respectively. in the literature for these materials. It predicts the disorder-induced asymmetry between the transition branches and correctly reproduces the scanning behavior or, more generally, any state within the sorption hysteresis loop. As proven by comparing the numerical solution of the mean field theory of lattice gas for a model set of channels with disorder and the analytical prediction of SCPM, the SCPM theory is found to be correct not only qualitatively, but quantitatively. In light of this, SCPM may be used to substantially improve the structural analysis methods based on the measurements of the phase transition points. The theory presented is exemplified for capillary condensation and evaporation phenomena, but

23



Figure 6: Sorption scanning loops obtained using MFT (symbols) for the disordered pores and the predictions of SCPM (solid lines). a shows a loop with the closure point on the adsorption boundary curve, while b shows a loop with the closure point on the desorption boundary. is easily extended to other transitions, such as melting and freezing. Moreover, it can also be reformulated for phenomena apart from first-order phase transitions, like mercury intrusion and extrusion. In all these cases, only the respective transformations of the microscopic transition kernel is needed. In future work we will explore to what extend the theory formulated and solved for linear chains of pores can be applied to describe phase equilibria in real porous solids with more complex pore networks rather than MCM-41-like. Indeed, some transition pathways can be dominated by the percolation phenomena and the interconnectivity of the pore network under this circumstance is of crucial importance. For example, the capillary desorption transition in the pressure range above that where cavitation is effective is known to be controlled by gas invasion percolation. Hence, this transition may not be reproduced by SCPM for materials with networked pore spaces. On the other hand, as far as random materials with sufficiently small pore sizes to trigger cavitation in the pore body are considered, nucleation of gas bubble will effectively facilitate the phase transformation by providing many gas phase growth points within the material. The experimental evidence for overcoming the limitations posed by the invasion percolation character of the desorption transition and the

24


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applicability of SCPM under these conditions was demonstrated recently. 55 In contrast, for the capillary condensation transition, nucleation of the liquid bridges is an effective source for germinating phase growth and the SCPM results can accurately capture the behavior of a variety of porous solids. Another aspect, which will be addressed in future, is related to the fact that SCPM yields not only the average phase compositions, namely the transition isotherms, but also the lengths distributions of different phases along the pore spaces. Thus, the SCPM theory offer the possibility to address physical phenomena, which intrinsically depend on these distributions. Among them, correlating the light scattering patterns with phase equilibria in porous solids is perhaps the most interesting application of the SCPM theory.

Acknowledgement The German Science Foundation (DFG) is gratefully acknowledged for the financial support (Projects No. 249197121 and 411771259)

Appendixes Appendix I. Independent pore limit (L = 1) By considering the adsorption boundary curve, this section exemplifies that the statistical SCPM theory in the limit of L = 1 transforms to the general adsorption equation, Eq. (1)). For L = 1, all pore sections are independent and all domain sizes are restricted to λ = 1. Consequently, all domain length distributions (thus also Eq. (10)) can be rewritten to

L=1

fg (λ, p) −−→

   1 λ = 1

.

  0 λ 6= 1

25


(53)


Page 26 of 34

Eqs. (12) and (14) then amount to

Pst (p) =

Pn Pg

Ptr (p) = 0.

(54)

With that, PSD of the filled pore sections (Eq. (5)) can be expressed simply as PSD with cut-off at xn : ψ(x) = φ ×

   1 x ≤ xn

.

(55)

  0 xn < x Inserting Eq. (55) into the SCPM adsorption isotherm, Eq. (3), gives (ads) θSCPM (p)

Z

Z

=

Z

θg φdx + x≤xn

θn φdx =

(ads)

θn φdx = θGAI (p).

(56)

x>xn

In Eq. (56) it was utilized that the kernels for nucleation and growth coincide in the region, where the pores are definitively filled because the thermodynamic conditions allow for nucleation, i.e. θn (x ≤ xn (p), p) = θg (x ≤ xn (p), p) ∀ p.

Appendix II. Long pore limit (L → ∞) For mesoporous samples with very large grain sizes strong disorder, SCPM in the limit of infinitely long pores can be a reasonable approximation. In this regime, the resulting sorption and scanning isotherms can be simplified as presented in this section using the example of the adsorption boundary curve. For large L, the length distribution of domains consisting of only sections with x ≤ xg (p), see Eq. (10), can be written as

L→∞

fg (λ, p) −−−→

   1

Pg = 1, λ = ∞

  Pgλ−1 (1 − Pg ) otherwise

26


.

(57)

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The mean probability to have access to a nucleus, Eq. (12), is L→∞

Pst (p) −−−→ 1 −

Pg − Pn (1 − Pg )2 . Pg (1 − Pg + Pn )2

(58)

Using Eqs. (14), (5) and (3), the adsorption boundary curve for L 1 can be approximated as (ads) θSCPM (p)

Z =

Z [θg Ptr + θn (1 − Ptr )] φdx +

θg φdx + x≤xn

Z

xn

Capillary Condensation and Evaporation in Irregular Channels

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