Capillary Condensation and Evaporation in Irregular Channels ......of computational modeling using the mean ﬁeld theory of lattice gas in irregular model pores. The theory developed

Capillary Condensation and Evaporation in Irregular Channels:Sorption Isotherm for Serially Connected Pore ModelDaniel Schneider and Rustem Valiullin*

Felix Bloch Institute for Solid State Physics, University of Leipzig, Linneśtr. 5, 04103 Leipzig, Germany

ABSTRACT: Geometrical disorder can strongly impact phaseequilibria of fluids in mesoporous solids. There is insufficientknowledge of how the structural disorder results in the emergence ofthe cooperativity effects in phase transitions. To tackle this problem,understanding of the complex interplay between nucleation and phasegrowth and the pore space morphology is needed. We use statisticallydisordered chains of serially connected single pores with varying poresizes to mimic geometric disorder and solve the problem using astatistical thermodynamics approach. As the main result, we derive theexact solution for the average phase composition at any thermody-namic condition including all states within the hysteresis region, i.e.,the entire family of the sorption isotherms including the scanningisotherms. We show that our approach correctly reproduces the resultsof computational modeling using the mean field theory of lattice gas in irregular model pores. The theory developed is directlyapplicable to the analysis of phase equilibria in materials with tubular pores, such as MCM-41 and SBA-15, but can also be usedto gain deeper insight into phase behavior in mesoporous solids with random pore networks.

■ INTRODUCTIONUnderstanding phase equilibria of fluids confined in smallpores is of importance in many areas of science andtechnology, including chemical engineering, environmentaland applied sciences, medicine, etc. Some important examplesare freezing of water in porous dust particles as an inhibitor ofice formation in clouds,1 friction induced by capillarycondensation,2−4 sound and light propagation in partiallysaturated porous solids,5,6 optical switching by capillarycondensation,7 altering of heat transfer at nanoscale,8 andmaterial deformation due to capillary condensation.9 Manyaspects of these phenomena occurring in pore spaces withsimple pore morphologies, such as in cylindrical channels or inslit-like pores (referred to in what follows as single pores), haveextensively been addressed in the literature (see, for example,recent reviews10−15). Summarizing the most essential points ofthese studies, three important aspects may be discerned.Foremost, the phase coexistence lines for materials innanoscale pores are found to be shifted with respect to thatof bulk fluids. These shifts scale typically in proportion withthe inverse pore size. The second common observation is thatphase transitions of confined materials typically exhibitirreversibility. Finally, the boundary conditions at the poreopenings decide on the transition mechanism by eitherintroducing or by removing the nucleation barriers.In materials with complex pore morphologies, phase

equilibria become increasingly intricate.16−18 First of all, theemerging distribution of the confinement sizes leads to arespective distribution of the transition points, such as thecondensation pressures. For a collection of separated singlepores with a distribution of the pore sizes, the respective

spread of the transition points can easily be quantified. Thisbecomes possible due to the fact that the boundary conditionsat the pore openings of each pore in the ensemble are identicaland thus the overall transition behavior becomes simplycumulative of the behaviors in each single pore. Anyinterconnection between the single pores complicates theproblem notably. In this case, the transition mechanism in aselected pore becomes, in addition, determined by the phasestate in the adjacent pore. Convincing evidence for thisscenario has been provided by the studies of the so-called ink-bottle pore systems.19−25 In materials with the geometricdisorder, this coupling between phase states in different partsof the pore network gives rise to the strong cooperativecharacter of the phase transitions and, hence, to very complexphase equilibria. Theoretical description of phase transitionsunder these conditions is a challenging problem.26−34

In recent decades, there was growing evidence that, inmaterials with seemingly ideal single pore structures, the phasetransitions may exhibit some features typical for disorderedmaterials.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 beuntypical of single pore materials.35,38,44 Both these features,typically observed in mesoporous solids with complex poremorphologies, such as Vycor porous glass, so far wereattributed to network effects.17,33,45,46 Establishing the under-

Received: April 17, 2019Revised: June 4, 2019Published: June 7, 2019

Article

pubs.acs.org/JPCCCite This: J. Phys. Chem. C 2019, 123, 16239−16249

© 2019 American Chemical Society 16239 DOI: 10.1021/acs.jpcc.9b03626J. Phys. Chem. C 2019, 123, 16239−16249

Dow

nloa

ded

via

UN

IV L

EIP

ZIG

on

July

22,

201

9 at

06:

21:0

3 (U

TC

).Se

e ht

tps:

//pub

s.ac

s.or

g/sh

arin

ggui

delin

es f

or o

ptio

ns o

n ho

w to

legi

timat

ely

shar

e pu

blis

hed

artic

les.

pubs.acs.org/JPCChttp://pubs.acs.org/action/showCitFormats?doi=10.1021/acs.jpcc.9b03626http://dx.doi.org/10.1021/acs.jpcc.9b03626

lying reasons for their observations in materials with thetubular pore structure is thus essential for both validating andimproving theoretical models for phase equilibria on a singlepore level and for better understanding the real structure ofmaterials under study. One of the hypotheses suggested is thatcorrugations of the pore walls and undulations along thechannel axes can be responsible for the phenomenaobserved.41−43,47−49 Computer-based numerical analysesusing linear pores composed of cylindrical pore sections withvarying diameters indeed confirmed the emergence of thepatterns typical of real, geometrically disordered poroussolids.50−52

It was shown recently that, for statistically disordered linearchains of pores with varying pore sizes, in what follows referredto as serially connected pore model (SCPM), the phaseequilibria can be obtained analytically using a statisticalthermodynamics approach.34 In particular, an approximatesolution for the boundary phase transitions and the scanningtransitions were obtained. The sorption isotherms derivedagreed qualitatively well with the experimental observations.However, the approximation made had two essential draw-backs. First, the approximate isotherm did not allow for anaccurate quantitative analysis, especially for short chains ofsingle pores, which is the case in MCM-41 or SBA-15materials. It is worth noting that having an accurate solution isimportant not only from the material characterizationperspective, but also the accurate model for one-dimensionaldisordered chains provides a basis for quantitative analysis ofnetwork effects in real materials. Second, and mostimportantly, the earlier theory did not provide how differentphases, capillary-condensed and gaseous, are distributed alongthe pore space. The latter, as now provided in the presentwork, opens new avenues for addressing complex physicalphenomena depending on the phase distribution properties. Asthe most relevant examples, light scattering and mass transferin partially filled porous solids, which are still far from beingquantitatively described, may be mentioned. In the presentwork, we provide the details how the sorption isotherms can beobtained for the SCPM and prove the accuracy of the solutionobtained by numerical solution of the mean field theory forlattice gas in the model pore systems. Notably, the theoryoutlined here may easily be adopted for describing freezing andmelting transitions in porous materials, mercury intrusion andextrusion, and also related structural transitions exhibiting poresize-dependent properties. The derivation details may also beimportant for those aiming at analysis of scattering propertiesunder conditions of phase coexistence in mesoporous solids.

■ RESULTSPore Model and Transition Mechanisms. The pore

spaces in our work are represented by chains of joinedcylindrical pore sections with different diameters x and equallength l (see Figure 1), in what follows referred to as theserially connected pore model (SCPM). The section volumes

are distributed according to a normalized distribution functionϕ(x), the so-called pore size distribution (PSD). It is assumedthat the diameters x of the two adjacent channel sections arestatistically independent. The total length of a channel, L, ismeasured in units of l and thus depicts the number ofstatistically uncorrelated pore sections. During sorption inmesopores, vapor (gray) and condensed (blue) phases maycoexist, forming gaseous and liquid domains, i.e., continuousdomains containing either capillary-condensed liquid orgaseous phase coexisting with a liquid film on the channelwalls. The length of such phase domains in units of l is denotedby λ. Large ensembles of such channels with differentrealizations of the disorder are considered allowing for astatistical average.With the variation of the gas pressure, the phase equilibria in

the pores depend on the governing phase transitionmechanisms. It is considered that the phase compositionmay change by nucleation and by phase growth as explained inmore detail in what follows. As for the first-order phasetransitions, the lack of nuclei of the new phase may prevent thephase transition, trapping the old phase in a metastable state.In this case, a nucleation event is needed for the transition tooccur. For capillary condensation in pores, this mechanismcorresponds to the phenomenon called liquid bridging,whereas for evaporation it refers to cavitation. We assumethat as soon as a nucleus is formed in a pore section, phasetransition in the entire pore section is stimulated. In thefollowing, this mechanism is referred to by nucleation anddenoted with the subscript n. Phase transformation in somepore sections may also be triggered by an already existingnucleus. The role of a nucleus can be played by the liquid orgas phase already formed in a pore section directly adjacent tothe pore section in which phase transformation is considered.Alternatively, it may be the new phase supplied at the channelopenings. Because metastability, in this case, is removed, thesetransitions occur in thermodynamic equilibrium. The respec-tive phenomena in sorption are associated with advancedcondensation or evaporation via gas invasion. In the remainderof this work, this mechanism is termed growth and is indicatedwith the subscript g.Whether the capillary transition in an arbitrarily selected

pore section occurs by nucleation or growth is determined bythermodynamic conditions, such as temperature and gaspressure, the section diameter x, and the boundary conditions,that is, the phase state in the adjacent pore sections. The latterintroduces a nearest-neighbor coupling in the chain and leadsto the cooperativity effects in the phase transitions. The phasebehavior in separate, single pore sections is described by theso-called kernels, i.e., theoretically or experimentally obtainedsets of sorption isotherms for varying pressure p and differentpore diameters x.In the following, the two nucleation kernels are denoted with

θn(x,p) for liquid bridging and θn′(x,p) for cavitation. Becauseof reversibility in thermodynamic equilibrium, the growthkernels for advanced condensation and gas invasion coincide,i.e., θg(x,p) = θg′(x,p). From these kernels, the critical diametersfor each transition mechanism can also be obtained, dividingthe pore sections in terms of their pore sizes into two parts,that is, in which nucleation and growth may occur and inwhich not. These critical diameters are denoted with xn(p),xg(p) = xg′(p), and xn′(p) for liquid bridging, advancedcondensation/gas invasion, and cavitation, respectively. Forreadability, in what follows the condensation process and the

Figure 1. Schematic illustration of a discrete model to describestatistically disordered one-dimensional pores.

The Journal of Physical Chemistry C Article

DOI: 10.1021/acs.jpcc.9b03626J. Phys. Chem. C 2019, 123, 16239−16249

16240

http://dx.doi.org/10.1021/acs.jpcc.9b03626

pore sections containing the capillary-condensed phase aretermed as filling/filled, while evaporation and the pore sectionscontaining the gaseous phase coexisting with the physisorbedfilm on the channel walls are denoted as emptying/empty.Also, the quantities corresponding to emptying/empty arelabeled with the prime symbol (′).SCPM Adsorption Isotherm. Based on the independent

domain model (IDM), the general adsorption isotherm (GAI)equation is widely accepted to relate sorption in a disorderedporous solid to the kernels of the independent pore sections itis comprised of.53 For the adsorption boundary curve, it can beexpressed as

p x p x x( ) ( , ) ( )dGAIads

n∫θ θ ϕ= (1)The only transition mechanism considered by eq 1 incylindrical pores, as often assumed in the literature, isnucleation. Similarly, if a spherical pore approximation isused, metastability is inherently removed and only theequilibrium transition condition is considered. In this case,θn(x,p) is replaced by θg(x,p). Since IDM treats the phasebehaviors in all pore sections independently, stimulated phasegrowth along the chain becomes ineffective. Hence, eq 1underestimates 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 termto eq 1. In particular, one may write the SCPM isotherm foradsorption as

p x( ) ( ) dads GAI(ads)

g n∫θ θ θ θ ψ= + − (2)where ψ indicates PSD of the filled sections only. Equation 2can be modified to

p x x( ) d dads g n∫θ θ ψ θ ψ= + ′ (3)where ψ′(x,p) denotes PSD of the empty sections and ψ + ψ′ =ϕ. For desorption, the SCPM isotherm can be expressed in asimilar way as

p x x( ) d ddes n g∫θ θ ψ θ ψ= ′ + ′ ′ (4)Physically, eqs 3 and 4 divide the integral over the poresections into two parts. The first represents the contribution ofall filled pore sections, whereas the second one gives thecontribution of the empty sections amounting in just adsorbedlayers on the pore walls. In what follows, we first describe howto obtain PSDs ψ and ψ′ for the main adsorption anddesorption transitions. Note that, as soon as ψ and ψ′ areestablished, the phase equilibria become fully described.Adsorption Boundary Curve. For obtaining the adsorption

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

x p

x x

P x x x

x x

( , )

1

0

n

tr n g

g

l

moooooo

noooooo

ψ ϕ= ×

≤

< ≤

< (5)

PSD of the empty pore sections can simply be obtained by ψ′= ϕ − ψ. Equation 5 implies that filling of every pore sectionwith a diameter smaller than xn is triggered by nucleation. Forxn < x ≤ xg, only a certain fraction Ptr(p) of the pore sectionsare filled by growth from the adjacent sections already filledwith the condensed phase. Hence, Ptr(p) represents theprobability that a randomly selected pore section is connecteddirectly or indirectly, via other pore sections with x ≤ xg, to asection with the condensed liquid. Finally, all pore sectionswith x > xg remain empty because they cannot be filled byeither mechanism.In eq 5, Ptr is the sole unknown quantity. To obtain Ptr, we

start with introducing the mean probabilities Pn and Pg that anarbitrarily selected pore section has a sufficiently small size toallow for nucleation or growth, respectively. For cylindricalgeometry of the pore sections, both probabilities can beobtained as

P px x

x x( )

d

dx x

n

2

2n

∫

∫

ϕ

ϕ= ≤

−

−(6)

and

P px x

x x( )

d

d

x xg

2

2g

∫

∫

ϕ

ϕ=

≤−

−(7)

Note that these probabilities turn out to be the cumulativeprobabilities of normalized number distribution functions. Themean total probability that a section is filled by either of themechanisms is

P px x

x xP P P P( )

d

d( )

2

2 n tr g n∫∫

ψ

ϕ= = + −

−

−(8)

Alternatively, P can be expressed as

P p P P( ) g st= (9)

Equation 9 implies that two conditions need to be fulfilled for apore section to fill: (i) the section has to be narrow enough forcapillary condensation to be possible, i.e., x ≤ xg. This is simplyPg. (ii) The section may be either small enough for anucleation event to occur (x ≤ xn) or, otherwise, needs to haveaccess to at least one nucleus at its boundaries. The respective,combined probability is denoted by Pst (“stimulated”). Toobtain Pst, the length distribution fg(λ) of the continuousdomains composed of the pore sections meeting the condition(i) needs to be found. Here, λ denotes the length of thedomains in units of a number of the pore sections.Combinatorial analysis yields

f pP LP LP

P P L P L

P L

( , )1

(1 ) 2 ( 1)(1 )

L

gg2

g g2

g g g

g

lmoooo

noooo

λ

λ λ

λ

=+ −

− [ + − − − ] <

=

λ

(10)

Any such domain fulfilling the condition (i) can only be filled ifat least one nucleation event can occur within the domain, i.e.,it has to entail at least one pore section with x < xn. Thisnucleus will stimulate filling of the entire domain by growth.



16241


The corresponding probability p( , )st λ for a domain of lengthλ to be filled is

p fPP

PP

( , ) 1 0, , 1 1st binn

g

n

g

i

kjjjjjj

y

{zzzzzz

i

kjjjjjj

y

{zzzzzzλ λ= − = − −

λ

(11)

where f bin denotes the binominal distribution. With the help ofeqs 10 and 11, the mean probability Pst is found as an average

P p( ) ( ) fst st ( )gλ= ⟨ ⟩λ λ (12)

where the mean value operator is defined as

X yX y F y

F y( )

( ) ( )

( )F yy

y( )⟨ ⟩ =

∑

∑ (13)

By comparing eqs 8 and 9 one obtains

P pP P P

P P( )tr

st g n

g n=

−− (14)

With Ptr being obtained, PSDs of the filled and empty sectionsbecome fully determined. Ultimately, the adsorption boundarycurve can be obtained with eq 3. It is worth noting that, in thelimit of L = 1, the SCMP isotherm (eq 3) naturally convergesto that of IDM as given by eq 1. For proof, see the Appendix.To complement the description of the phase state in the

pore, the length distribution f(λ,p) of the filled domains can beobtained via

f pf

f( , )

( ) ( )

( ) ( )Lst g

1 st g

λλ λ

λ λ=

∑λ= (15)

Along with the average phase composition, as expressed by theadsorption isotherm, eq 15 provides the complete statisticaldescription of the phase equilibria along the adsorptionboundary 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, xn′(p) and xg′(p),respectively. Initially, all pore sections are filled with thecapillary-condensed liquid. Upon a quasistatic decrease of thegas pressure, the pore sections empty gradually. PSDassociated with the empty sections only is

x p

x x

P x x x

x x

( , )

0

1

g

tr g n

n

l

moooooo

noooooo

ψ ϕ′ = ×

≤ ′

′ ′ < ≤ ′

′ < (16)

where xn′(p) > xg′(p) ∀ p, and PSD of the filled sections is foundto be ψ = ϕ − ψ′. Every pore section larger than xn′ is emptiedby cavitation of gas bubbles. For the sections in the range xg′ <x ≤ xn′, only the fraction Ptr′ can be emptied by gas invasion,that is, only those sections in contact with the pore sections inthe empty state. All pore sections with x ≤ xg′ remain filled.Similar to adsorption, an expression for Ptr′ is the unknown tobe obtained. Similar to eqs 7 and 6, we introduce the meanprobabilities Pn′ and Pg′ as

P px x

x x( )

d

dx x

n

2

2n

∫

∫

ϕ

ϕ′ = > ′

−

−(17)

and

P px x

x x( )

d

d

x xg

2

2g

∫

∫

ϕ

ϕ′ =

> ′−

−(18)

Now, Ptr′ , PSDs of the empty and filled sections, and f ′(λ,p)can be obtained with essentially the same procedure as for theadsorption boundary curve by replacing all quantitiescorresponding to the filled phase with those of the emptyphase (X → X′). The only adjustment needs to be made isrelated to the fact that, at the channel openings, there is directcontact to the gas phase at all pressures. This provides anotherpathway for emptying in addition to that triggered bycavitation. With this, the total probability ( )st λ′ for a domainof length λ to be emptied is

p( , )st st,n st,g st,n st,g st,n st,gλ′ = ′ ∨ ′ = ′ + ′ − ′ ′(19)

The probability st,n′ for cavitation in the domain and theprobability st,g′ to have access to the external vapor phase are

pPP

( , ) 1 1st,nn

g

i

kjjjjjj

y

{zzzzzzλ′ = − −

′′

λ

(20)

and

pL P

( , )2

2 ( 1)(1 )st,g gλ

λ′ =

+ − − − ′ (21)

respectively. Ptr′ , the quantity completely determining thesolution of the problem, is given by

P pP P P

P P( )tr

st g n

g n′ =

′ ′ − ′′ − ′ (22)

with

P p p( ) ( , ) fst st ( )gλ′ = ⟨ ′ ⟩λ λ′ (23)

where fg′(λ) is given by eq 10 with Pg being replaced by Pg′.Finally, the desorption boundary curve can be obtained byinserting PSDs for the empty and filled sections into eq 4.

Desorption Scanning Curves. Let us now consider thestates obtained within the hysteresis loop, which are achieved,for example, by performing desorption upon incompleteadsorption. The desorption scanning curves can be obtainedwith essentially same consideration as the boundary isotherms,but with the correspondingly adjusted initial conditions. Let usdenote with p0 the pressure attained along the boundaryadsorption branch, where the desorption scan is initiated. Atthis point, complex distribution of the empty and filleddomains along the channels is found. The corresponding PSDsof the filled and empty pore sections at p0 are denoted withψ0(x) (eq 5) and ψ0′(x) = ϕ − ψ0, respectively. The lengths Λof the filled domains subject to emptying during the desorptionscan, LΛ ∈ ≤ , are distributed according to f 0(Λ), as given byeq 15. From this initial phase state, the desorption scan isperformed with decreasing pressure. At a pressure p < p0, PSDs



16242


of the filled and empty sections along the desorption scanningcurve can be obtained as

x p p( , , )0 0ψ ψ ψ= − Δ ′ (24)

x p p( , , )0 0ψ ψ ψ′ = ′ + Δ ′ (25)

where

x p p

x x

P x x x

x x

( , , )

0

10 0

g

tr g n

n

l

moooooo

noooooo

ψ ψΔ ′ = ×

≤ ′

′ ′ < ≤ ′

′ < (26)

In the equations above, only the mean fraction of the poresections emptied by growth, Ptr′ is unknown. To derive Ptr′ , themean probabilities for an arbitrary pore section to empty bynucleation and growth are needed. In the spirit of eqs 17 and18, they are found as

P p px x

x x( , )

d

dx x

n 0

20

20

n∫

∫

ψ

ψ′ = > ′

−

−(27)

P p px x

x x( , )

d

d

x xg 0

20

20

g∫

∫

ψ

ψ′ =

> ′−

−(28)

The mean total probability that a section is emptied by eitherone of the mechanisms is found as

P p px x

x xP P P P( , )

d

d( )0

2

20

n tr g n∫∫

ψ

ψ′ =

Δ ′= ′ + ′ ′ − ′

−

−(29)

where

P p pP P P

P P( , )tr 0

st g n

g n′ =

′ ′ − ′′ − ′ (30)

(see, for the derivation, the discussion preceding eq 14).To obtain Pst′ in eq 30, the length distribution of the

continuous domains which can be formed by gas invasionneeds to be found. For this purpose, desorption in each of theinitially filled domains drawn from f 0(Λ) needs to be treatedindividually. In such a domain of length Λ, the lengths λ ≤ Λof the smaller domains that may empty during the desorptionscan, are distributed as

f p pP P P

P P P

P

( , , , )1

(1 ) 2 ( 1)(1 )

g 0g

2g g

2

g g g

g

lmoooo

noooo

λ

λ λ

λ

′ Λ =′ + Λ ′ − Λ ′

×′ − ′ [ + Λ − − − ′ ] < Λ

′ = Λ

λ

Λ

(31)

For these domains to empty, at least one nucleation eventwithin the domain has to occur. The probability of this event is

st,n′ . Alternatively, it can empty if contact to the empty phaseis provided at the boundary. The respective probability is st,g′ .The combined probability is then

p p( , , , )st 0 st,n st,g

st,n st,g st,n st,g

λ′ Λ = ′ ∨ ′

= ′ + ′ − ′ ′ (32)

With respect to eq 32, the probability for at least onenucleation event to occur is calculated as

p pPP

( , , ) 1 1st,n 0n

g

i

kjjjjjj

y

{zzzzzzλ′ = − −

′′

λ

(33)

The probability for contact with an adjacent empty phase canbe obtained with

p pP

( , , , )2

2 ( 1)(1 )st,g 0 gλ

λ′ Λ =

+ Λ − − − ′ (34)

Finally, the mean probability Pst′ (p,p0) is found as

P p p( , ) ( , ) f fst 0 st ( , ) ( )g 0λ′ = ⟨⟨ ′ Λ ⟩ ⟩λ λ′ Λ Λ Λ (35)

By inserting eqs 35 and 30 into eq 26 PSDs of the filled andempty pore sections become fully determined. The desorptionscanning curve can now be obtained using eq 4. Additionally,the length distribution of the empty domains formed duringthe scan from p0 to p can be obtained with

f p pf

f( , , )

( , ) ( , )

( , ) ( , )Lf

0

st g

1 st g ( )0

λλ λ

λ λ′ =

′ Λ ′ Λ

∑ ′ Λ ′ Λλ=Λ Λ (36)

Adsorption Scanning Curves. A similar strategy is usedfor the derivation of the adsorption scanning curves. Consideran adsorption scan starting at pressure p0 on the desorptionboundary curve. Let us denote PSDs of the empty and filledpore sections at this initial point with ψ0′(x), as given by eq 16,and ψ0(x) = ϕ − ψ0′, respectively. The lengths LΛ ∈ ≤ of theempty domains, subject to filling during the subsequentpressure increase, are distributed according to f 0′(Λ). Thisdistribution function is given by eq 15, but with correspond-ingly replaced parameters relevant for desorption. At a pressurep > p0, PSDs of the filled and empty pore sections can beobtained as

x p p( , , )0 0ψ ψ ψ= + Δ (37)

x p p( , , )0 0ψ ψ ψ′ = ′ − Δ (38)

where

x p p

x x

P x x x

x x

( , , )

1

00 0

n

tr n g

g

l

moooooo

noooooo

ψ ψΔ = ′ ×

≤

< ≤

< (39)

Once again, to calculate Ptr, we introduce the meanprobabilities for an arbitrary pore section to fill by nucleationand growth as

P p px x

x x( , )

d

dx x

n 0

20

20

n∫

∫

ψ

ψ=

′

′≤

−

−(40)

P p px x

x x( , )

d

d

x xg 0

20

20

g∫

∫

ψ

ψ=

′

′≤

−

−(41)

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



16243


P p pP P P

P P( , )tr 0

st g n

g n=

−− (42)

To obtain Pst, the length distribution of the continuousdomains found in the empty state and fulfilling the condition x≤ xg needs to be found. For this purpose, adsorption in each ofthe initially empty domains drawn from f 0′(Λ) may be treatedindividually. In such a domain of length Λ, the lengths λ ≤ Λof the domains we are looking for are distributed as

f p pP P P

P P P

P

( , , , )1

(1 ) 2 ( 1)(1 )

g 0g2

g g2

g g g

g

lmoooo

noooo

λ

λ λ

λ

Λ =+ Λ − Λ

×− [ + Λ − − − ] < Λ

= Λ

λ

Λ

(43)

In these domains, at least one nucleus is needed to fill theentire domain with the condensed liquid. There are twoscenarios how this can be accomplished: either a liquid bridgecan be formed within a domain (the respective probability is

st,n) or direct contact to a filled phase domain at the boundaryis given (the respective probability is st,g). The combinedprobability is then

p p( , , , )st 0 st,n st,g st,n st,g st,n st,gλ Λ = ∨ = + −(44)

The probability for at least one nucleation event to occur is

p pPP

( , , ) 1 1st,n 0n

g

i

kjjjjjj

y

{zzzzzzλ = − −

λ

(45)

The probability of having contact with an adjacent filledsection is

p pb

P( , , , )

22 ( 1)(1 )st,g 0

0

gλ

λΛ =

+ Λ − − − (46)

where b0 accounts for the existence of a phase interface at theboundary of the domains to be filled as follows. In the interiorof a channel with the alternating filled and empty domains,each empty domain is in contact with the filled domains onboth sides. However, an empty domain at either of the channelends is in contact with only one filled domain due to theexternal bulk phase on the other side. Thus, with n0′ indicatingthe mean number of the empty domains in a channel at p0, wedefine b0 as b0 = (n0′ − 1)/n0′. n0′ itself can be found by therelationship between the combined length of the emptydomains and the mean length of one empty domain, i.e., n0′ =P0′L/⟨Λ⟩. Hence, b0 can be expressed as

bP L L

x x

x xf1 1

1 d

d( )d

L

00

2

20 1

0∑∫∫

ϕ

ϕ= − ⟨Λ⟩

′= −

′Λ ′ Λ Λ

−

−Λ=

(47)

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

P p p( , ) ( , ) f fst 0 st ( , ) ( )g 0λ= ⟨⟨ Λ ⟩ ⟩λ λ Λ Λ ′ Λ (48)

Inserting eqs 48 and 42 into eq 39 fully determines PSDs ofthe filled and empty sections, and the adsorption scanningcurve is obtained using eq 3. Additionally, the lengthdistribution of the filled domains formed during the scan canbe obtained with

Figure 2. (a) Sorption isotherms exemplifying liquid bridging (solid black line), advanced adsorption and gas invasion (dashed red line), andcavitation (dotted blue line) as obtained with mean field theory for an ideal cylindrical pore with a diameter of 10 nm. The other figures show thekernels for liquid bridging (θn(x,p), (b), advanced adsorption and desorption (θg(x,p) = θg′(x,p), (c), and cavitation (θn′(x,p), (d). In (b)−(d), thedifferent curves represent different pore diameters x, starting with 3 nm in the upper left corner and increasing to 28 nm toward the bottom rightcorner. All curves were obtained at T/Tc = 3/4.



16244


f p pf

f( , , )

( , ) ( , )

( , ) ( , )Lf

0

st g

1 st g ( )0

λλ λ

λ λ=

Λ Λ

∑ Λ Λλ=Λ ′ Λ (49)

■ DISCUSSIONTo verify SCPM, we have used the mean field theory of thelattice gas model to explore the phase equilibria in longdisordered pores generated using computer-based algorithms.In particular, long channels consisting of L = 500interconnected cylindrical pore sections with their diametersdrawn randomly from a γ distribution with the mean μ = 12nm, shape factor k = 6, and two cut-offs taken at 3 and 28 nmwere generated. Note that the diameters drawn werediscretized and varied in steps of 1 nm. Each of the poresections had a length of 25 nm, which gives a total channellength of 12.5 μm. To obtain the sorption isotherms, the meanfield theory was used. For a more accurate statisticalrepresentation, the results were averaged over differentchannels with different disorder realizations. Because ofextremely long computations, only an average of over 8sample pores was managed.The simple cubic lattice gas model in an external field with

nearest-neighbor interactions was employed to describe thephase behavior in the thus obtained channels following thework by Woo et al.54 The configurational energy wascalculated as

H n n n n( )2i ij

i ji

i i∑ ∑ε ϕ= − +

⟨ ⟩ (50)

where ni represents the occupation number on a latticecoordinate vector i, ⟨ij⟩ denotes the sum over all nearest-neighbor pairs, ε is the intermolecular interaction constant, andϕi models interaction of molecules with the surface. The grandcanonical potential in the mean-field approximation is

kT

2

1ln (1 )ln(1 )

iji j

ii i

ii

ii i i i

∑ ∑ ∑

∑

ε ρ ρ ρ ϕ μ ρ

ρ ρ ρ ρ

Ω = − + −

+ [ + − − ]

⟨ ⟩

(51)

where ρ = ⟨ni⟩ is the ensemble average and μ is the chemicalpotential related to the gas pressure. By minimizing Ω, theequilibrium densities at each T, V, and μ are found as

kT1 exp

1i

jj i

1i

k

jjjjjjjj

lmooonooo

|}ooo~ooo

y

{

zzzzzzzz∑ρ ε ρ ϕ μ= + [− + − ]

−

(52)

Sorption Kernels. First, the mean field theory was appliedto obtain the respective kernels for the phase transitionmechanisms. For this purpose, ideal cylindrical channels ofdifferent diameters were used. Figure 2a exemplifies the kernelisotherms for all phase transition mechanisms obtained for apore of 10 nm diameter at T/Tc = 3/4. To obtain the kernelisotherms for liquid bridging, θn(x,p), open-ended pores incontact with a bulk gas phase on both sides were used. Sincethis morphology lacks a nucleus, during adsorption, capillarycondensation can only originate from the nucleation of a liquidbridge. The advanced adsorption and desorption kernels,θg(x,p) and θg′(x,p), were acquired with the capped poregeometry closed by a pore wall on one side and open toward a

bulk gas phase on the other. Here, the transitions occur viagrowth from a nucleus, either the capped end for condensationor the open end for evaporation. Condensation andevaporation via growth are reversible, i.e., θg = θg′. Here, thecapped end pore was used as an approximation of a situation,where an adjacent pore section contains the capillary-condensed liquid. The cavitation kernel, θn′(x,p), was obtainedupon desorption with a closed pore geometry, makingintrusion of the gas phase impossible. Thus, here the onlymechanism to evaporate is the nucleation of a gas bubble.Figure 2b−d show all kernels obtained. Note that surface layeradsorption is observed in the lower pressure regime. For allisotherms shown, two distinct states with a sharp transition inbetween can be seen, the gaseous state with adsorbed surfacelayers (empty) and the capillary-condensed state (filled) forpressures close to the saturated vapor pressure. Figure 3 showsthe critical pore diameters for all phase transition mechanisms,that is, xn for liquid bridging, xg = xg′ for advanced adsorptionand gas invasion, and xn′ for cavitation.

Transitions in Disordered Channels. Figure 4 shows theMFT results obtained with the disordered channels, in whichthe pore section diameters are distributed according to the γdistribution as described earlier. A relatively broad hysteresisloop is formed primarily due to strong pore blocking. Thisbecomes evident by comparing the MFT results to theprediction of the IDM theory shown by the lines. The latter

Figure 3. Critical pore diameters for the different phase transitionmechanisms upon variation of the gas pressure at T/Tc = 3/4. Theblack dots indicate the critical diameters for liquid bridging (xn), thered triangles for advanced adsorption and gas invasion (xg = xg′), andthe blue diamonds for cavitation (xn′).

Figure 4. Adsorption and desorption boundary curves obtained fromthe MFT calculation for the disordered pores (symbols) and aspredicted by the independent domain model (solid lines).



16245


predicts significantly narrower hysteresis between adsorptionand desorption and fails to reproduce both transition branches.In contrast to IDM, as demonstrated in Figure 5a, SCPM

shows almost perfect match between theory and the MFTresults (note that the step-like behavior seen in the theoreticalpredictions, results from a discrete set of the kernels and thesection diameters used). Moreover, the SCPM theoryreproduces perfectly not only the boundary transitions butalso the scanning behavior. This is exemplified in Figure 5showing also the desorption scans obtained using MFT andSCPM. Additional proof of the validity of SCPM isdemonstrated in Figure 6 evidencing that SCPM is capableof reproducing any state within the hysteresis loop obtained,for example, for more complex partial isotherms, such asscanning loops.

■ CONCLUSIONSIn the present work we present statistical thermodynamicstheory for phase equilibria in irregular one-dimensional pores.Without any rigorous restrictions it can be applied to describegas sorption in porous solids with unidirectional pores, such asMCM-41 or SBA-15. By directly incorporating the disordereffects, modeled by variation of the channel diameter along thechannel axes, and by allowing for the interplay betweennucleation and phase growth, the SCPM theory is found toreproduce the majority of the experimental findings reported inthe literature for these materials. It predicts the disorder-induced asymmetry between the transition branches andcorrectly reproduces the scanning behavior or, more generally,any state within the sorption hysteresis loop. As proven bycomparing the numerical solution of the mean field theory oflattice gas for a model set of channels with disorder and theanalytical prediction of SCPM, the SCPM theory is found to

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, c) Desorption and adsorption scanning curves obtained using MFTfor the disordered pores and the predictions of SCPM (solid lines), respectively.

Figure 6. Sorption scanning loops obtained using MFT (symbols) for the disordered pores and the predictions of SCPM (solid lines). (a) Shows aloop with the closure point on the adsorption boundary curve, while (b) shows a loop with the closure point on the desorption boundary.



16246


be correct not only qualitatively, but quantitatively. In light ofthis, SCPM may be used to substantially improve the structuralanalysis methods based on the measurements of the phasetransition points. The theory presented is exemplified forcapillary condensation, and evaporation phenomena but iseasily extended to other transitions, such as melting andfreezing. Moreover, it can also be reformulated for phenomenaapart from first-order phase transitions, like mercury intrusionand extrusion. In all of these cases, only the respectivetransformations of the microscopic transition kernel areneeded.In future work, we will explore to what extent the theory

formulated and solved for linear chains of pores can be appliedto describe phase equilibria in real porous solids with morecomplex pore networks rather than MCM-41-like. Indeed,some transition pathways can be dominated by the percolationphenomena, and the interconnectivity of the pore networkunder this circumstance is of crucial importance. For example,the capillary desorption transition in the pressure range abovethis, where cavitation is effective, is known to be controlled bygas invasion percolation. Hence, this transition may not bereproduced by SCPM for materials with networked porespaces. On the other hand, as far as random materials withsufficiently small pore sizes to trigger cavitation in the porebody are considered, nucleation of the gas bubble willeffectively facilitate the phase transformation by providingmany gas phase growth points within the material. Theexperimental evidence for overcoming the limitations posed bythe invasion percolation character of the desorption transitionand the applicability of SCPM under these conditions wasdemonstrated recently.55 In contrast, for the capillarycondensation transition, nucleation of the liquid bridges is aneffective source for germinating phase growth, and the SCPMresults can accurately capture the behavior of a variety ofporous solids. Another aspect, which will be addressed infuture, is related to the fact that SCPM yields not only theaverage phase compositions, such as the transition isotherms,but also the length distributions of different phases along thepore spaces. Thus, the SCPM theory offer the possibility toaddress physical phenomena, which intrinsically depend onthese distributions. Among them, correlating the lightscattering patterns with phase equilibria in porous solids isperhaps the most interesting application of the SCPM theory.

■ APPENDIXESAppendix I. Independent Pore Limit (L = 1)By considering the adsorption boundary curve, this sectionexemplifies that the statistical SCPM theory in the limit of L =1 transforms into the general adsorption equation, eq 1. For L= 1, all pore sections are independent and all domain sizes arerestricted to λ = 1. Consequently, all domain lengthdistributions (thus, also eq 10) can be rewritten as

f p( , )1 10 1

L

g

1 lmoonoo

λλλ

⎯ →⎯⎯=≠

=

(53)

Equations 12 and 14 then amount to

P pPP

( )stn

g=

P p( ) 0tr = (54)

With this, PSD of the filled pore sections (eq 5) can beexpressed simply as PSD with cut-off at xn

xx x

x x( )

1

0n

n

lmoonoo

ψ ϕ= ×≤< (55)

Inserting eq 55 into the SCPM adsorption isotherm, eq 3, gives

p x x x p( ) d d d ( )x x x x

SCPM(ads)

g n n GAI(ads)

n n∫ ∫ ∫θ θ ϕ θ ϕ θ ϕ θ= + = =

≤ >

(56)

In eq 56, it was utilized that the kernels for nucleation andgrowth coincide in the region, where the pores are definitivelyfilled because the thermodynamic conditions allow fornucleation, 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 strongdisorder, SCPM in the limit of infinitely long pores can be areasonable approximation. In this regime, the resultingsorption and scanning isotherms can be simplified as presentedin this section using the example of the adsorption boundarycurve. For large L, the length distribution of domainsconsisting of only sections with x ≤ xg(p), see eq 10, can bewritten as

f pP

P P( , )

1 1,

(1 ) otherwise

L

g

g

g1

g

lmoooonooo

λλ

⎯ →⎯⎯⎯⎯= = ∞

−λ→∞

−(57)

The mean probability to have access to a nucleus, eq 12, is

P pP P

P

P

P P( ) 1

(1 )

(1 )L

stg n

g

g2

g n2⎯ →⎯⎯⎯⎯ −

− −

− +→∞

(58)

Using eqs 14, 5, and 3, the adsorption boundary curve for L ≫1 can be approximated as

p x P P

x x

( ) d (1 )

d d

x x x x x

x x

SCPM(ads)

g g tr n tr

n

n n g

g

∫ ∫

∫

θ θ ϕ θ θ

ϕ θ ϕ

= + [ + − ]

+

≤ < ≤

< (59)

with

P pP

P P( ) 1

(1 )

(1 )trg

2

g n2= −

−

− + (60)

where Pn and Pg refer to eqs 6 and 7, respectively. Onconsidering the distribution of the domain lengths, that for thefilled domains is obtained as

f p

P

P

P P P

PP P P

( , )

0 0

1 1,

(1 )(1 )

( )

otherwise

L

n

g

g g n

n

g g n

l

m

oooooooooooo

n

oooooooooooo

λ

λ

⎯ →⎯⎯⎯⎯

=

= = ∞

− − +

× [ − − ]λ λ

→∞

(61)

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected].



16247

mailto:[email protected]://dx.doi.org/10.1021/acs.jpcc.9b03626

ORCIDRustem Valiullin: 0000-0001-5028-7642NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThe German Science Foundation (DFG) is gratefullyacknowledged for the financial support (Projects Nos249197121 and 411771259).

■ REFERENCES(1) Campbell, J. M.; Christenson, H. K. Nucleation- and Emergence-Limited Growth of Ice from Pores. Phys. Rev. Lett. 2018, 120,No. 165701.(2) Jinesh, K. B.; Frenken, J. W. M. Capillary Condensation inAtomic Scale Friction: How Water Acts like a Glue. Phys. Rev. Lett.2006, 96, No. 166103.(3) Feiler, A.; Stiernstedt, J.; Theander, K.; Jenkins, P.; Rutland, M.Effect of Capillary Condensation on Friction Force and Adhesion.Langmuir 2007, 23, 517−522.(4) Greiner, C.; Felts, J. R.; Dai, Z. T.; King, W. P.; Carpick, R. W.Controlling Nanoscale Friction through the Competition betweenCapillary Adsorption and Thermally Activated Sliding. ACS Nano2012, 6, 4305−4313.(5) Page, J. H.; Liu, J.; Abeles, B.; Herbolzheimer, E.; Deckman, H.W.; Weitz, D. A. Adsorption and Desorption of a Wetting Fluid inVycor Studied by Acoustic and Optical Techniques. Phys. Rev. E1995, 52, 2763−2777.(6) Ogawa, S.; Nakamura, J. Hysteretic Characteristics of 1/Lambda(4) Scattering of Light During Adsorption and Desorption ofWater in Porous Vycor Glass with Nanopores. J. Opt. Soc. Am. A2013, 30, 2079−2089.(7) Barthelemy, P.; Ghulinyan, M.; Gaburro, Z.; Toninelli, C.;Pavesi, L.; Wiersma, D. S. Optical Switching by CapillaryCondensation. Nat. Photonics 2007, 1, 172−175.(8) Holyst, R.; Litniewski, M. Heat Transfer at the Nanoscale:Evaporation of Nanodroplets. Phys. Rev. Lett. 2008, 100, No. 055701.(9) Günther, G.; Prass, J.; Paris, O.; Schoen, M. Novel Insights intoNanopore Deformation Caused by Capillary Condensation. Phys. Rev.Lett. 2008, 101, 086104−4.(10) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; Sliwinska-Bartkowiak, M. Phase Separation in Confined Systems. Rep. Prog.Phys. 1999, 62, 1573−1659.(11) Monson, P. A. Understanding Adsorption/DesorptionHysteresis for Fluids in Mesoporous Materials using Simple MolecularModels and Classical Density Functional Theory. MicroporousMesoporous Mater. 2012, 160, 47−66.(12) Landers, J.; Gor, G. Y.; Neimark, A. V. Density FunctionalTheory Methods for Characterization of Porous Materials. ColloidsSurf., A 2013, 437, 3−32.(13) Coasne, B.; Galarneau, A.; Pellenq, R. J. M.; Di Renzo, F.Adsorption, Intrusion and Freezing in Porous Silica: The View fromthe Nanoscale. Chem. Soc. Rev. 2013, 42, 4141−4171.(14) Jiang, Q.; Ward, M. D. Crystallization under NanoscaleConfinement. Chem. Soc. Rev. 2014, 43, 2066−2079.(15) Huber, P. Soft matter in hard confinement: phase transitionthermodynamics, structure, texture, diffusion and flow in nanoporousmedia. J. Phys.: Condens. Matter 2015, 27, No. 103102.(16) Everett, D. H. In The Solid−Gas Interface; Alison Flood, E., Ed.;Marcel Dekker, Inc.: New York, 1967; pp 1055−1113.(17) Burgess, C. G. V.; Everett, D. H.; Nuttall, S. AdsorptionHysteresis in Porous Materials. Pure Appl. Chem. 1989, 61, 1845−1852.(18) Rigby, S. P. Recent (Developments in the Structural)Characterisation of Disordered, Mesoporous Solids. Johnson MattheyTechnol. Rev. 2018, 62, 296−312.

(19) Ravikovitch, P.; Neimark, A. Experimental Confirmation ofDifferent Mechanisms of Evaporation from Ink-Bottle Type Pores:Equilibrium, Pore Blocking, and Cavitation. Langmuir 2002, 18,9830−9837.(20) Libby, B.; Monson, P. A. Adsorption/Desorption Hysteresis inInkbottle Pores: A Density Functional Theory and Monte CarloSimulation Study. Langmuir 2004, 20, 4289−4294.(21) Morishige, K.; Yasunaga, H.; Denoyel, R.; Wernert, V. Pore-Blocking-Controlled Freezing of Water in Cagelike Pores of KIT-5. J.Phys. Chem. C 2007, 111, 9488−9495.(22) Khokhlov, A.; Valiullin, R.; Kar̈ger, J.; Steinbach, F.; Feldhoff,A. Freezing and Melting Transitions of Liquids in Mesopores withInk-Bottle Geometry. New J. Phys. 2007, 9, 272.(23) Casanova, F.; Chiang, C. E.; Li, C.-P.; Schuller, I. K. DirectObservation of Cooperative Effects in Capillary Condensation: TheHysteretic Origin. Appl. Phys. Lett. 2007, 91, 243103−3.(24) Bruschi, L.; Mistura, G.; Liu, L.; Lee, W.; Gosele, U.; Coasne,B. Capillary Condensation and Evaporation in Alumina Nanoporeswith Controlled Modulations. Langmuir 2010, 26, 11894−11898.(25) Zeng, Y.; Tan, S. J.; Do, D. D.; Nicholson, D. Hysteresis andCurves in Linear Arrays of Mesopores with Two Cavities and ThreeNecks: Classification of the Curves. Colloids Surf., A 2016, 496, 52−62.(26) Mason, G. A Model of Adsorption Desorption Hysteresis inWhich Hysteresis is Primarily Developed by the Iinterconnections in aNetwork of Pores. Proc. R. Soc. A 1983, 390, 47−72.(27) Neimark, A. V. In Studies in Surface Science and Catalysis;Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Elsevier,1991; Vol. 62, pp 67−74.(28) Liu, H.; Zhang, L.; Seaton, N. A. Analysis of SorptionHysteresis in Mesoporous Solids Using a Pore Network Model. J.Colloid Interface Sci. 1993, 156, 285−293.(29) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Sarkisov, L.;Tarjus, G. Capillary Condensation in Disordered Porous Materials:Hysteresis versus Equilibrium Behavior. Phys. Rev. Lett. 2001, 87,No. 055701.(30) Cordero, S.; Rojas, F.; Kornhauser, I.; Dominguez, A.; Vidales,A. M.; Lopez, R.; Zgrablich, G.; Riccardo, J. L. Pore-Blocking andPore-Assisting Factors during Capillary Condensation and Evapo-ration. Appl. Surf. Sci. 2002, 196, 224−238.(31) Šooś,̌ M.; Rajniak, P.; Stepanek, F. Percolation Models ofAdsorption-Desorption Equilibria and Kinetics for Systems withHysteresis. Colloids Surf., A 2007, 300, 191−203.(32) Handford, T. P.; Perez-Reche, F. J.; Taraskin, S. N. CapillaryCondensation in One-Dimensional Irregular Confinement. Phys. Rev.E 2013, 88, No. 012144.(33) Cimino, R.; Cychosz, K. A.; Thommes, M.; Neimark, A. V.Experimental and Theoretical Studies of Scanning Adsorption-Desorption Isotherms. Colloids Surf., A 2013, 437, 76−89.(34) Schneider, D.; Kondrashova, D.; Valiullin, R. Phase Transitionsin Disordered Mesoporous Solids. Sci. Rep. 2017, 7, No. 7216.(35) Kruk, M.; Jaroniec, M.; Sayari, A. Nitrogen Adsorption Study ofMCM-41 Molecular Sieves Synthesized using Hydrothermal Re-structuring. Adsorption 2000, 6, 47−51.(36) Coasne, B.; Grosman, A.; Ortega, C.; Simon, M. Adsorption inNoninterconnected Pores Open at One or at Both Ends: AReconsideration of the Origin of the Hysteresis Phenomenon. Phys.Rev. Lett. 2002, 88, No. 256102.(37) Wallacher, D.; Kunzner, N.; Kovalev, D.; Knorr, N.; Knorr, K.Capillary Condensation in Linear Mesopores of Different Shape. Phys.Rev. Lett. 2004, 92, No. 195704.(38) Tompsett, G. A.; Krogh, L.; Griffin, D. W.; Conner, W. C.Hysteresis and Scanning Behavior of Mesoporous Molecular Sieves.Langmuir 2005, 21, 8214−8225.(39) Coasne, B.; Hung, F.; Pellenq, R.-M.; Siperstein, F.; Gubbins,K. Adsorption of Simple Gases in MCM-41 Materials: The Role ofSurface Roughness. Langmuir 2006, 22, 194−202.



16248

http://orcid.org/0000-0001-5028-7642http://dx.doi.org/10.1021/acs.jpcc.9b03626

(40) Petrov, O.; Furo, I. A Study of Freezing-Melting Hysteresis ofWater in Different Porous Materials. Part II: Surfactant-TemplatedSilicas. Phys. Chem. Chem. Phys. 2011, 13, 16358−16365.(41) Gommes, C. J. Adsorption, Capillary Bridge Formation, andCavitation in SBA-15 Corrugated Mesopores: A Derjaguin-Broekhoff-de Boer Analysis. Langmuir 2012, 28, 5101−5115.(42) Morishige, K. Nature of Adsorption Hysteresis in CylindricalPores: Effect of Pore Corrugation. J. Phys. Chem. C 2016, 120,22508−22514.(43) Guillet-Nicolas, R.; Berube, F.; Thommes, M.; Janicke, M. T.;Kleitz, F. Selectively Tuned Pore Condensation and HysteresisBehavior in Mesoporous SBA-15 Silica: Correlating MaterialSynthesis to Advanced Gas Adsorption Analysis. J. Phys. Chem. C2017, 121, 24505−24526.(44) Esparza, J. M.; Ojeda, M. L.; Campero, A.; Dominguez, A.;Kornhauser, I.; Rojas, F.; Vidales, A. M.; Lopez, R. H.; Zgrablich, G.N-2 Sorption Scanning Behavior of SBA-15 Porous Substrates.Colloids Surf., A 2004, 241, 35−45.(45) Kondrashova, D.; Reichenbach, C.; Valiullin, R. Probing PoreConnectivity in Random Porous Materials by Scanning Freezing andMelting Experiments. Langmuir 2010, 26, 6380−6385.(46) Petrov, O.; Furo, I. A Study of Freezing-Melting Hysteresis ofWater in Different Porous Materials. Part I: Porous Silica Glasses.Microporous Mesoporous Mater. 2011, 138, 221−227.(47) Liu, J.; Shin, Y.; Nie, Z. M.; Chang, J. H.; Wang, L. Q.; Fryxell,G. E.; Samuels, W. D.; Exarhos, G. J. Molecular Assembly in OrderedMesoporosity: A New Class of Highly Functional NanoscaleMaterials. J. Phys. Chem. A 2000, 104, 8328−8339.(48) Puibasset, J. Monte-Carlo Multiscale Simulation Study ofArgon Adsorption/Desorption Hysteresis in Mesoporous Heteroge-neous Tubular Pores like MCM-41 or Oxidized Porous Silicon.Langmuir 2009, 25, 903−911.(49) Morishige, K. Effects of Carbon Coating and Pore Corrugationon Capillary Condensation of Nitrogen in SBA-15 Mesoporous Silica.Langmuir 2013, 29, 11915−11923.(50) Coasne, B.; Gubbins, K. E.; Pellenq, R. J.-M. Domain Theoryfor Capillary Condensation Hysteresis. Phys. Rev. B 2005, 72,No. 024304.(51) Kondrashova, D.; Valiullin, R. Freezing and Melting Transitionsunder Mesoscalic Confinement: Application of the Kossel-StranskiCrystal-Growth Model. J. Phys. Chem. C 2015, 119, 4312−4323.(52) Morishige, K. Dependent Domain Model of Cylindrical Pores.J. Phys. Chem. C 2017, 121, 5099−5107.(53) Thommes, M.; Kaneko, K.; Neimark, A. V.; Olivier, J. P.;Rodriguez-Reinoso, F.; Rouquerol, J.; Sing, K. S. W. Physisorption ofGases, with Special Reference to the Evaluation of Surface Area andPore Size Distribution (IUPAC Technical Report). Pure Appl. Chem.2015, 87, 1051−1069.(54) Woo, H. J.; Sarkisov, L.; Monson, P. A. Mean-Field Theory ofFluid Adsorption in a Porous Glass. Langmuir 2001, 17, 7472−7475.(55) Enninful, R.; Schneider, D.; Hoppe, A.; König, S.; Fröba, M.;Enke, D.; Valiullin, R. Comparative Gas Sorption and CryoporometryStudy of Mesoporous Glass Structure: Application of the SeriallyConnected Pore Model. Front. Chem. 2019, DOI: 10.3389/fchem.2019.00230.



16249
http://dx.doi.org/10.3389/fchem.2019.00230http://dx.doi.org/10.3389/fchem.2019.00230http://dx.doi.org/10.1021/acs.jpcc.9b03626

Documents

Capillary Condensation and Evaporation in Irregular Channels ......of computational modeling using the mean ﬁeld theory of lattice gas in irregular model pores. The theory developed