Unexpected Sorption-Induced Deformation of Nanoporous Glass

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Unexpected sorption-induced deformation of nanoporous glass: evidence for spatial rearrangement of adsorbed argon Klaus Schappert, and Rolf Pelster Langmuir, Just Accepted Manuscript • DOI: 10.1021/la502974w • Publication Date (Web): 30 Oct 2014 Downloaded from http://pubs.acs.org on November 13, 2014

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Unexpected sorption-induced deformation of nanoporous glass: evidence for spatial rearrangement of adsorbed argon Klaus Schappert∗ and Rolf Pelster∗ FR 7.2 Experimentalphysik, Universität des Saarlandes, 66123 Saarbrücken, Germany

E-mail: [email protected]; [email protected]

Abstract Sorption of substances in pores generally results in a deformation of the porous matrix. The clarication of this eect is of particular importance for the recovery of methane and the geological storage of CO2 . As a model system we study the macroR glass during sorption of argon using capaciscopic deformation of nanoporous Vycor

tative measurements of the length change of the sample. Upon desorption we observe an unpredicted sharp contraction and re-expansion peak which contains information on the draining mechanism of the porous sample. We have modied the theoretical model by Gor and Neimark 1 to predict the sorption-induced deformation of (partly) lled porous samples. In this analysis the contraction is attributed to a metastable or non-equilibrium conguration where a thin surface layer on the pore walls coexists with capillary bridges. Alternatively, pore blocking and cavitation during the draining of the polydisperse pore network can be at the origin of the deformation peak. The results are a substantial step towards a correlation between the spatial conguration of adsorbate, its interaction with the host material, and the resulting deformation. ∗

To whom correspondence should be addressed 1

ACS Paragon Plus Environment

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Introduction Sorption of substances in porous materials or stones is an ubiquitous phenomenon in nature. Water, for instance, is adsorbed in sandstone or other natural stone but it can also be imbibed in the pores of concrete and cause damages. Of current social and economic interest is the recovery of natural gas from porous rocks and the geological storage of CO 2 .

25

But also in

the automobile industry porous materials are considered for the storage of methane.

6

For all

these applications it is highly important to know how the porous material reacts upon the adsorption or desorption of a substance. It is well-established that sorption inuences and in particular deforms porous matrices (see e.g. Refs.

1,2,4,5,723,25

). Both the curvature of the

interface of the adsorbate and changes of the surface free energy are related to a pressure on the porous matrix that causes this deformation.

1

Also the impact of attractive van der

Waals forces in small cavities, the electrical properties of the adsorbate, and the inuence of dispersion forces on the deformation behavior were studied.

7,21,24

The deformation behavior

for small pore llings of less than one monolayer is rather complex and both an expansion or a contraction can be observed. properties of adsorbates.

24

24

This behavior was attributed to dierences in the electrical

But the adsorption of a few layers of adsorbate on the pore walls

of porous samples with a three-dimensional network of interconnected pores mostly causes a marked extension (see e.g. Ref.,

15,19

an exception being water in activated charcoal

22

). In

contrast, for materials with parallel pores the rst adsorbed surface layers usually cause a signicantly smaller deformation than capillary bridges at high lling.

7,18,21

The extent of the sorption-induced deformation depends decisively on the elastic properties of the porous system.

1,9

Up-to-date, however, no denite clarication of the correlation

between the elastic properties of the porous matrix, the pore lling, and the deformation could be achieved. Several formulas were established to estimate the elastic properties of the porous material (assuming dierent structures of the network) from the measured sorptioninduced deformation.

12,15,18,19,21,23

But these equations mostly supply elastic properties that

are of the right order of magnitude only.

12,15,18,19,21

For the theoretical description of the

2


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deformation it is usually assumed that the elastic moduli are independent of the pore lling.

9

Beside this correlation problem also the geometrical arrangement of the pore lling and

the shape of the adsorbate's interface play a decisive role for the deformation of the porous matrix. For the understanding of these inuences, we thoroughly study in this article the behavior of a simple model system. We investigate argon in nanoporous glass (average pore radius

rP ≈ 3.8 − 4.0 nm) with the aid of ultrasonic measurements, capacitative deformation

measurements, and theoretical considerations.

Experimental details The porous matrix used for the measurements is the well-known Vycor Inc.).

26

R

glass (Corning

It possesses a spongy three-dimensional structure of interconnected pores.

average pore radius

rP

27

The

of the samples used in the measurements varies slightly between

≈ 3.8 nm and ≈ 4.0 nm (porosity φ ≈ 0.25−0.28).

The size distribution of the pore radius for

one sample is shown in Fig. 1. Assuming cylindrical pores these radii were caculated from the desorption branch of an isotherm with liquid argon as the sum of the radius of withdrawing menisci (using the Kelvin equation) and the thickness pore wall (empirical Halsey equation;

t ≈ 0.9

nm).

t of the remaining surface layer on the

2830

This method may give a somewhat

distorted impression of the actual size distribution because of the occurance of pore blocking during desorption and the assumption of strictly cylindrical pores.

28,31

The fraction of bigger

pores is possibly underestimated and the fraction of smaller pores overestimated because of the pore blocking eect.

28

The sample cells are mounted on the cold head of a cryostat and are surrounded by an insulation vacuum ( p

. 5 · 10−6

mbar) for the measurements at the required low tempera-

tures (cp. schematic sketch in Fig. 2). For the regulation of the temperature each sample cell includes a temperature sensor (silicon diode) and a heating resistor that are connected to a temperature controller. This makes it possible to stabilize the temperature to the re-

3


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Figure 1: (Color online) The porous Vycor

R

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glass exhibits a pore size distribution.

The

draining mechanism may inuence the form of the shown distribution (see text).

quired value (uctuations smaller than temperature is

±0.25

±10

mK). The absolute accuracy of the measured

K.

Figure 2: Sketch of an experimental setup with cold head and gas distribution system.

The lling of the sample with argon is varied by dened changes of the pressure

p

of the

argon gas in the gas distribution system that is connected to the sample cell via a capillary (cp. schematic sketch in Fig. 2). The molar amount value for the reduced pressure

p/p0

n

of adsorbate in the pores at a certain

(with the bulk vapor pressure of argon,

4


p0 ) is determined

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volumetrically with the aid of the ideal gas equation.

f = n/n0 = VAr,ads /Vpores complete lling,

Vpores .

n0 ,

32

Thus, the volume lling fraction

can be evaluated, with the molar amount of argon necessary for

the volume of the adsorbed argon,

(A lling fraction of

VAr,ads ,

f > 1 at p/p0 = 1 corresponds

and the volume of the pores,

to the formation of condensate in

the sample cell.) Sorption dynamics in porous materials are rather complex.

p

Therefore, after each

p several hours were necessary to reach a stable value for the

change of the external pressure pressure

33,34

and the length of the sample. At the end of the paper we will discuss whether

always a stable equilibrium state and thus a stable conguration of the adsorbed argon was reached. For the measurement of a complete sorption isotherm we needed up to two months. The eective longitudinal modulus

β = c2l ρ of the heterogeneous system of porous matrix

and pore lling is determined using the ultrasonic velocity frequency

νl ≈ 12

M:

of longitudinal waves (carrier

MHz) propagating through the sample (pulse echo method, cp. Ref.

and the eective density empty sample,

cl

ρ = ρ0 (1 + nm/M )

38

)

of the (partly) lled sample ( ρ0 : density of the

mass of the empty sample;

m:

molar mass of argon).

pulses are generated by applying voltage pulses to a piezo-crystal (LiNbO 3 ,

The ultrasonic

36◦ Y-cut;

both

sides coated with a thin gold lm) that is glued with a silver epoxy composite on the sample (see Fig. 3). The length change of the porous sample during lling is measured with a capacitative distance sensor.

This sensor is mounted a few micrometers above the sample onto which

a gold surface was sputtered. change of the capacity. Refs.

15,19

Any change in the length of the sample thus reects in a

(Such a capacitative measurement method was, e.g., also used in

) For the deformation measurements we used two samples with dierent lengths

but very similar average pore radii ( rP empty sample was l0

= 2.41

≈ 3.8 − 4.0

nm, see above): The initial length of the

mm (for the measurements at

(for the measurement at

86

maximum length change

∆lmax

K). We estimate the error in at

p/p0 = 1).

80

K and

∆l/∆lmax

74

K) or l0

to be

= 2.64

≤ 0.05

mm

(with the

A schematic sketch of the sample cell for the

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Figure 3: (Color online) Schematic sketch of a sample cell for the ultrasonic measurements. The copper sample cell is mounted on the cold head of a cryostat (cp. Fig. 2).

deformation measurements is shown in Fig. 4.

Results and Discussion Deformation during sorption of argon and the eective longitudinal modulus Sorption of argon results in a complex deformation behavior of the nanoporous Vycor sample. The change of the sample's length

R

glass

∆l depends signicantly on whether the sample is

lled or emptied. Equal amounts of adsorbate can result in both an expansion or contraction of the pores and the sample (cp. Figs. 5a+b). The observed deformation mainly resembles what is known from literature (for other systems), except the observed very pronounced contraction and re-expansion at a specic value of the reduced vapor pressure desorption (see Fig. 5b at

p/p0 ≈ 0.72).

p/p0

during

Note that we have done the sorption measurements

very slowly: after each change of the lling of the sample the system required several hours to reach a stable state.

6


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Figure 4: (Color online) Schematic sketch of the sample cell for the deformation measurements. The copper sample cell is mounted on the cold head of a cryostat (cp. Fig. 2).

After adding some liquid argon at a temperature of up to a reduced pressure in the range of

p/p0 ≈ 0.8

86 K, the sample considerably expands

(see Fig. 5b). This pressure corresponds

to the steep part of the adsorption branch (see Fig. 5a). It can be assumed that roughly up to this value the adsorbate forms surface layers on the pore walls. The measured expansion is mainly caused by the reduction of the surface free energy and the related pressure is exerted by the liquid layer on the pore walls. The Laplace pressure adsorbed lm of thickness

t

1,8

−γlv / (rP − t)

that

of the

counteracts this expansion ( γlv : liquid-vapor surface tension).

Its contribution is, however, comparatively small for thin lms. At higher external pressure

1

1

p additional lling forms capillary bridges in the pores.

These

become energetically more favorable at rst in smaller pores, while in bigger pores the thickness of the surface layer still increases. Thus there are two coexistent congurations of the adsorbate in the pores. Now the negative Laplace pressure in both capillary bridges with concave menisci and adsorbed surface layers seeks a reduction of the pore diameter and of the size of the sample. But the increasing thickness of the surface layer in regions without

1

capillary bridges leads to a further reduction in the surface free energy (cp. Ref. ). Thus, there is once again an interplay between the two opposite deformation eects. But above

7


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Figure 5: (Color online) Inuence of adsorbed liquid argon in nanoporous glass. (a) Volume

f = VAr,ads /Vpores of the pores with argon (86 K) as a function of the reduced pressure p/p0 (p0 : vapor pressure of bulk argon). Below p/p0 ≈ 0.8 the adsorbate

lling fraction vapor

forms surface layers on the pore wall. At higher pressure in the steep part of the isotherm (p/p0

≈ 0.8)

capillary bridges are formed that vanish during desorption at

The adsorbed argon results in a deformation at

p/p0 = 1).

∆l

of the sample (scaled to

p/p0 ≈ 0.7. (b) ∆lmax = 1.0 µm

The circle marks an unusual contraction and re-expansion that has not been

predicted theoretically. (c) The eective longitudinal modulus increases/decreases stepwise when the pores are completely lled/emptied ( β0

= 17.68

GPa).

(d) The increase of the

attenuation of the ultrasonic signal shows the appearance of vapor voids in the range of the formation and vanishing of capillary bridges.

8


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p/p0 ≈ 0.8

the contracting eect of the Laplace pressure predominates and a slight relative

contraction is observed up to

p/p0 ≈ 0.85,

where

lled with capillary bridges (see Figs. 5a+b).

f ≈1

holds and the pores are completely

There only remain concave menisci at the

pore ends. A subsequent increase of the pressure (cp. Fig. 5a) leads to a attening of those menisci and no perceptible increase of the lling; thus the negative pressure in the adsorbate is reduced and the sample expands. During desorption mainly the described behavior is reversed (see Fig. 5b). A reduction of the pressure in the sample cell yields to an increasing curvature of the menisci at the pore ends and hence a contraction of the sample is observed. In the steep desorption branch of the isotherm (see Fig. 5a) number and thickness of capillary bridges decrease which should lead to a continuous expansion of the sample until only surface layers exist. But, initially a sharp contraction of the sample is observed at

p/p0 ≈ 0.72

(see the circle in Fig. 5b). All of

our measurements with the globular argon show this contraction peak, however, it has not previously been observed in other systems, e.g. for water in porous glass.

19

At lower values of the reduced pressure the disappearance of capillary bridges becomes noticable by an expansion of the sample until only surface layers remain, just as it is expected (p/p0

≈ 0.65).

When the thickness of the surface layer, respectively the number of adsorbed

argon layers, is reduced in the further desorption process, the length reduces continuously to the value for the empty sample because of the related change of the surface free energy (cp. above). The comparison of the measured deformation with the eective longitudinal modulus and the attenuation of the ultrasonic pulses (see Fig. 5c+d) shows that local minima of both deformation curves (adsorption and desorption) correspond to stepwise changes of the eective longitudinal modulus. Above a reduced pressure of

p/p0 ≈ 0.8 during adsorption the

attenuation of the ultrasonic signal increases because of the appearance of capillary bridges and vapor voids, which has also been observed for hexane in Vycor

R . 34,39 Just before the

complete lling of the pores, when all pores are lled with capillary bridges, the eective

9


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modulus is increased in comparison to the empty sample. Adsorbed liquid surface layers do not result in a change of the eective modulus due to the existence of argon vapor in the pore center and the unrestricted mobility of the adsorbed atoms. At high reduced pressures ( p/p0

35

≥ 0.85 during adsorption and p/p0 ≥ 0.72 during desorp-

tion) both the eective longitudinal modulus

β

and the length change

∆l

are approximately

linear functions of the reduced pressure. The formation of concave menisci with decreasing radius

rc

pressure

at the pore ends during desorption leads to an increase of the negative Laplace

pL = 2γlv /rc

and thus both to a considerable contraction of the sample and to a

pressure-induced change of the adsorbate's intrinsic elastic properties.

25

The relationship between the deformation behavior and the eective longitudinal modulus is basically the same for all measured temperatures, i.e. in the liquid regime for 86 K and 80 K and in the solid regime for 74 K where the capillary condensate is crystalline and only surface layers at the pore walls remain liquidlike

40

(see Fig. 6). The sharp contraction peak

during desorption is dierently strong (cp. Fig. 5b at Fig. 6c at

p/p0 ≈ 0.75).

p/p0 ≈ 0.72, Fig. 6a at p/p0 ≈ 0.74, and

The inset in Fig. 6c shows that the extent of the sharp contraction

can vary for measurements with dierent samples at one and the same temperature. These dierences are mainly related to dierences in the geometrical structure of the pore lling during desorption. A dierent geometrical arrangement of the adsorbate can appear because of dierences in the pore structure of samples, in the size of individual desorption steps (with

∆pi ,

respectively

∆fi )

i

for one and the same sample, and because of (related) dierent

metastable or unequilibrated states of the adsorbate.

The signicance of the geometrical

structure of the lling will be thoroughly studied in the following sections. At a temperature of 74 K only a fraction of the lling is solid; the rst layers near the pore wall remain liquidlike and only at llings above argon exists.

38,40

f ≈ 0.66,

p/p0 ≈ 0.95

Therefore, the deformation behavior up to

dier from the liquid regime. But also above

p/p0 ≈ 0.95,

i.e. for

p/p0 ≈ 0.95,

solid

should mainly not

when solid adsorbate exists no

major dierences in the deformation behavior compared to the situation at

10


80

K and

86

K

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Figure 6:

(Color online) Scaled eective longitudinal modulus and deformation during

sorption of liquid or solid argon in nanoporous glass. Scaled length change of the nanoporous R glass as a function of the reduced vapor pressure p/p during sorption of argon at

Vycor 0 (a) 80 K, i.e. in the liquid regime (with regime (with

∆lmax = 0.9 µm).

∆lmax = 1.2 µm)

and (c) 74 K, i.e. in the solid

The circles mark a contraction peak that has not been

predicted theoretically. The longitudinal modulus increases at complete lling of the sample [(b) at 80 K (with

β0 = 16.88

β0 = 16.96 GPa)]. Because of increase of β is bigger at 74 K.

GPa) and (d) at 74 K (with

the higher longitudinal modulus of solid argon, the steep

As a result of the increase of the Laplace pressure during the initial desorption process at complete lling, the modulus decreases with decreasing external pressure (c) shows a desorption measurement for a dierent sample at

11


74

K.

p. 25

The inset in

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are observed. From the comparison of the sharp contraction of the sample during desorption at

0.72

(at

86

p/p0 ≈

K) to the eective longitudinal modulus one might conclude that the sharp

contraction is related to the observed weakening of the eective longitudinal modulus (see Fig. 5c) as a result of the appearance of vapor voids which becomes noticable in an increase of the attenuation of the ultrasonic signal (see Fig. 5d). In contrast, during adsorption at the steep increase of the longitudinal modulus near

p/p0 ≈ 0.85

no real peak in the deformation

is observed. At this higher reduced pressure the Laplace pressure is of course lower, but only by a factor of two in comparison to desorption. Consequently, the almost non-existence of a contraction peak during adsorption seems to contradict the impression that the steep change of the longitudinal modulus and the contraction peak might be related. At least the change of the modulus ( 5

− 10%)

is not the sole origin of the strong contraction peak.

For the further investigation of the origin of the strong contraction peak during desorption we rst discuss the theoretical model by Gor and Neimark

1

for the theoretical description of

the deformation behavior. Finally, their idealized model is extended to include the coexistence of adsorbed lms and capillary bridges, a condition that is typically present in porous samples over a certain pressure range.

Model by Gor and Neimark Gor and Neimark

1

used a thermodynamical model to describe the deformation at dierent

reduced pressures for cylindrical pores without any variation of the radius and without interconnections.

In this model the deformation behavior is described as a result of the

so-called solvation pressure

ps

that consists of pressure contributions for dierent llings.

The extent of the deformation sample and

∆Vpores

mechanical stress

σs

ε = ∆Vpores /V0

(with

V0

being the volume of the empty

the change of the pore volume) depends on the dierence between the induced by the adsorbate and the external pressure

usually negligible small.

1

p,

the later being

(Note that in experiments the total volume change of the porous

12


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sample is measured.) Assuming the validity of Hooke's law, the solvation pressure is related to the deformation

ε

via an elastic modulus of the pore system,

ε= The strain

ps + ε0 M

(1)

ε0 is an oset due to a prestress of the porous matrix.

modulus of the pore system,

9

M : 1,9

The modulus

M

is an elastic

but its meaning in terms of the moduli of the heterogeneous

system of porous sample and pore lling is not obvious. Measured eective moduli, e.g. the the longitudinal modulus

β,

relate an applied external pressure to an overall volume change

of a sample. In contrast, the modulus

M

in Eq. 1 relates the internal pressure exerted on the

pore walls to the overall volume change. Therefore, both quantities should not be confused. The models for the description of the deformation behavior regard the elastic properties of the sample usually as independent of the lling of the pores, the experimental observations (see Fig. 5 and e.g.

Refs.

34,35,40,41

).

9

which contrasts to

Discrepances between

the theoretical predictions and the measured deformation resulted in the assumption of a possible inuence of the eective elastic properties on the deformation behavior. independence of the modulus porous silica.

M

9

But the

in Eq. 1 on the pore lling corresponds to observations for

23

For completely lled pores with only concave menisci at the pore ends, the solvation pressure, i.e. the pressure exerted by the pore lling on the pore walls, is governed by the Laplace pressure.

23

The relation of the radius of curvature at dierent external pressures

can be calculated with the Kelvin equation, thus that the solvation pressure lled with capillary bridges is given by

pcs (p) = −

Vm ,

for pores

1

RT γwl + (p0 − p) + ln(p/p0 ), rP Vm

with the pore wall-liquid surface tension molar volume of the adsorbate

pcs

p

γwl ,

the gas constant

R,

(2)

the temperature

and the bulk vapor pressure of the adsorbate,

13


T,

p0 .

the The

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rst summand is a constant and the changes in the second are small ( ∆p

. 1 bar) compared

to the Laplace pressure (the third term). If there is solely an adsorbed lm on the pore walls the Laplace pressure of the adsorbed lm with the thickness

t

has to be considered as well as the altered surface energy and the

following relation holds for the solvation pressure

pfs (t) where

γlv

pfs

for adsorbed surface layers:

γlv t 1 γwl −p− − Π(t) − =− rP rP − t rP rP

is the liquid-vapor surface tension and

Π(t)

Z

∞

Π(t0 )dt0 ,

(3)

t

is the disjoining pressure which is for

argon-silica usually approximated with the Frenkel-Halsey-Hill (FHH) equation:

Π(t) = with the constants

t˜ = 0.1

nm,

k = 73.17

1

1

RT k Vm (t/t˜)m

and

m = 2.665

(4)

for argon (at

T = 87.3

K).

8

The rst term on the right hand side of Eq. 3 is the same constant as in Eq. 2 and the contribution of the pressure

−γlv /(rP − t)

p

is again very small.

The contracting Laplace pressure

of the adsorbed lm increases with increasing thickness

t.

The last two terms

with the disjoining pressure account for changes in the surface free energy in consequence of the adsorption of molecules on the pore walls. The absolute value of this free energy

1

contribution decreases with increasing lm thickness (cp. Ref. ). Thus there are mainly two competing deformation pressures for adsorbed lms. The model developed by Gor and Neimark

1

(on the basis of the Derjaguin-Broekho-de

Boer theory) allows to calculate the deformation behavior during sorption with the above equations.

42

The model presumes that at a certain pressure the pores ll abruptly and

completely with capillary bridges, i.e. it does not describe the coexistence of both adsorbate states, which is usually present in porous systems with a certain pore size distribution. The calculated solvation pressure during adsorption and desorption of liquid argon in pores with a pore radius of

rP = 4

nm is shown in Fig. 7 for reduced pressures above

14


p/p0 = 0.05

(for

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low pressures the FHH equation is not valid). sorption (see Eq. 1) as well as

Figure 7:

Assuming a constant modulus

∆Vpores ∝ ∆Vsample ,

be identic with the deformation typical volume changes with

43

M

during

the form of the solvation pressure should

εexp = ∆Vsample /V0 ≈ 3∆l/l0

(the latter equality holds for

∆Vsample 0.

If it is

assumed that in both cases the thickness of the adsorbed layer remains at a constant value

t0

above

fc

(see Figs. 8a+c), this thickness

f0 = 1 − (1 − t0 /rP )2 .

Above

f − f0 ξc = , 1 − f0

fc

t0

corresponds according to Eq. 6 to a lling

the pore lling is then

with

f = f0 · (1 − ξc ) + 1 · ξc

2 t0 f0 = 1 − 1 − rP

Eq. 7 permits accordingly the determination of the fraction capillary bridges above

fc ,

(for f ≥ fc ). ξc

16

(7)

of pore sections lled with

the only free parameter being the thickness


and thus

t0

of the surface lm.

Page 17 of 31

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Langmuir

Figure 8:

(Color online) (a)+(c) Thickness

appropriate fraction

ξc

t

of an adsorbed surface layer and (b)+(d)

of pore sections lled with capillary bridges as a function of the

lling during desorption. It is assumed that above a critical lling capillary bridges and surface layers coexist; below

fc

fc ≈ 0.47 (p/p0 ≈ 0.69)

there are (almost) no capillary bridges.

In (a)+(b) it is presumed that the thickness of the layer does not change abruptly at (t0

= 1.08

nm corresponding to 3 surface layers).

fc

The case of a rearrangement of the

adsorbate and the sudden change of the thickness of the layer is shown in (c)+(d) [capillary bridges become unstable and form surface layers at layer above

fc

fc ].

Here the thickness of the surface

is taken to be approximately the thickness of one surface layer,

17


u ≈ 0.36

nm.

Langmuir

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

In Fig. 8b+d the fraction

ξc

Page 18 of 31

of capillary bridges is shown for the two thickness dependences

from Fig. 8a+c. Here, it is assumed that during desorption almost all capillary bridges have disappeared at

p/p0 ≈ 0.69 and fc = fcdes ≈ 0.47 (see Fig. 5a; this lling fraction corresponds

to three surface layers). In the following, we use the isothermal sorption measurements (see Fig. 5a) to determine the reduced pressure at a certain lling pressure

pfs +c

f

and calculate the solvation

of a porous sample lled with both surface layers and capillary bridges.

In

doing so we assume certain congurations of the adsorbate, e.g. those shown in Fig. 8a+c for the desorption branch.

Figure 9:

(Color online) Solvation pressure during desorption assuming the coexistence of

adsorbed layers and capillary bridges for three dierent thickness dependences.

Only for

a thickness of the adsorbed layer corresponding to one surface layer ( t0 = 0.36 nm) above fcdes ≈ 0.47, the theoretical curve shows a sharp contraction peak that is qualitatively similar to the measured deformation.

The results of the calculations (according to Eq. 5) for three from Fig. 8a+c) are shown in Fig. 9. Assuming that Fig. 8a), the behavior of the solvation pressure

pfs +c

t

t0 -values

(including those

does not change abruptly at

fc

(cp.

is basically the same as in the simplied

case without the coexistence of surface layers and capillary bridges (cp. Figs. 7 and 9 for

t0 = 1.08 fcdes .

nm, i.e. three surface layers). The major dierence is only a broadening around

But for a thickness of

t0 = 0.36

nm (one surface layer) above

fcdes

(cp. Fig. 8c), the

solvation pressure shows a relatively sharp contraction peak and thus closely resembles the measured deformation during desorption (see Figs. 5b and 6a, peak widths are comparable,

18


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Langmuir

p/p0 -axes).

note the dierent scaling of the

Thus, for thickness dependences that are similar

to those shown in Fig. 8c one or even less surface layers , the theoretical curve corresponds well to the measurements. [Theoretically it is also possible that there is no lm on the pore walls above

fcdes , i.e. t0 = 0 nm.

in Fig. 9 for

t0 = 0.36

This would cause an even stronger peak than the one shown

t0 = 0

nm. (The solvation pressure for a lm tickness of

nm cannot

be calculated with Eq. 3, it tends to negative innity.)] At the end of this section we shall discuss whether the formation of such thin surface layers is possible and corresponds to the thermodynamical equilibrium. Note that in Eq. 3 we use the macroscopic Laplace equation along with the value

γlv

for

the macroscopic surface tension. But thin surface layers adsorbed on the walls of nanopores may exhibit dierent values. In addition, for the calculation of the disjoining pressure (see Eq. 4) we used the literature values for argon on silica. But, as it is generally known, the surface of porous Vycor

R

glass exhibits silanol OH-groups.

27

The possible deviations of the

used constants and the surface tension from the actual values in the porous system can increase or reduce the size of the contraction peak for a given layer thickness also shift the possible values of

t0

t0

and hence

slightly.

Now, we analyse the deformation behavior during the adsorption process. During adsorption the onset of capillary condensation can be at relatively high llings (above (cp. Fig. 5a). Therefore, a higher lling fraction,

fcads ≈ 0.59

p/p0 ≈ 0.76)

(four monolayers), must be

used for the calculation of the solvation pressure. The reduced pressure that corresponds to

fcads

is consequently also higher than during desorption. This shift of

fc

inuences also the

solvation pressure which is shown in Fig. 10 for four dierent thicknesses

t0

above

fcads .

In

contrast to the desorption branch, the best agreement with the measurements is achieved if there is no step in the thickness of the adsorbed layer (and the thickness remains constant at the onset of capillary condensation) or if the thickness monolayers) at

fcads

(cp. Figs. 5b and 10). Also

t0 = 0.72

t

jumps to

t0 = 1.08

nm (three

nm (two monolayers) is possible

because this thickness results in only a very small contraction near

19


fcads .

Langmuir

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

Figure 10:

Page 20 of 31

(Color online) Solvation pressure during adsorption assuming the coexistence

of adsorbed layers and capillary bridges for four dierent thickness dependences. During f +c adsorption the solvation pressure ps ts best to the measured deformation curve if there is no step in the thickness of the adsorbed layer or if the thickness t jumps to t0 = 1.08 nm at fcads ≈ 0.59 (cp. Fig. 5b). A qualitative comparison with the measured deformation suggests that at least the rst two surface layers remain stable when capillary bridges form.

Thus, when capillary bridges form during adsorption (at

fcads ≈ 0.59)

the surface lm

consisting of 4 monolayers may remain stable or reduces to 2 or 3 monolayers according to the analysis (cp. Fig. 10). The latter case would be more realistic for a monodisperse pore system, since the material needed for the capillary bridges has to come from the surface lms. When there is a distribution of pore sizes, also newly added material above

fc

may

form little by little bridges starting in the smallest pores. From theoretical considerations it is known that there is a stability limit for the thickness of adsorbed lms and consequently during adsorption the adsorbate forms capillary bridges above a metastable limit for the lm.

44,45

The calculations shown in Fig. 10 show that the third and fourth layer can be in a

metastable state, but clearly at least the rst two surface layers of argon are stable. The above analysis indicated the formation of a very thin surface layer (at maximum one monolayer) on the pore walls during the emptying of the pores in the steep part of the desorption branch.

But our sorption measurements (see Fig. 5a) indicate that the vapor

pressure of one monolayer ( f

≈ 0.17)

is

≈ 0.25 · p0 ,

≈ 0.69·p0 where the capillary bridges vanish.

i.e.

much less than the pressure of

Assuming that the individual measuring points

20


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Langmuir

during both adsorption and desorption represent stable equilibrium states, the thickness of the adsorbed layer during desorption at

p/p0 ≈ 0.69

should equal the thickness of the

adsorbed layer during adsorption at the same pressure. According to Eq. 6 the equilibrium thickness of the surface layer at

p/p0 ≈ 0.69

is

t ≈ 1.8 − 2

monolayers, i.e.

most pore

sections are covered with two argon monolayers and only a smaller fraction of sections are covered with one monolayer in contrast to the above analysis. But during desorption such a conguration would result in a smaller peak than the one shown in Fig. 9 for one monolayer (or no peak at all). Does this put our analysis into question? First, equalibration during sorption in porous systems is complex. Despite the very long waiting times between each measuring point (lling or emptying step) and the constancy of the reduced pressure it is not clear whether always a stable equilibrium was reached. In particular, it is possible that the conguration of the adsorbate causing the contraction peak during the emptying of the pores is metastable or even out of equilibrium.

It is known

that there are many possible metastable states for adsorbate in porous systems such as Vycor

R

glass.

4648

measurements.

Metastable states can also explain the existence of a hysteresis in sorption

4447

At

86

K we waited

22.5

hours after the rst lling step from

f ≈ 0.1 (corresponding to less than one surface layer).

f =0

But during desorption the pressure

to

p

and the length of the sample seemed to have reached a stable state in shorter time intervals: e.g.

≈3

hours for

f ≈ 1 → f ≈ 0.9.

after the step from

f ≈ 0.9

equilibrium thickness of

to

t0 ≈ 2

The minimum of the deformation curve was observed

f ≈ 0.8

and a waiting time of

monolayers at

p/p0 ≈ 0.69

16.5

hours.

Thus, even an

during adsorption cannot exclude

a thinner surface layer in the steep part of the desorption branch assuming metastable or unequilibrated states. Second, the Gor and Neimark model applies for monodisperse independent pores, but not necessarily for a network of interconnected polydisperse pores. A pore size distribution is known to inuence the draining mechanism of porous samples.

4951

Constrictions , i.e.

pore sections with smaller pore diameter, lead to the draining of the sample via the pore

21


Langmuir

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

blocking eect or cavitation.

49,50,52

Page 22 of 31

In the case of the pore blocking eect, pore sections with

smaller pore radius can prevent (block) the draining of pores with bigger radius, because a small pore will only empty at a lower pressure according to the Kelvin equation.

53

This

mechanism is known to result in a coarsening of the spatial distribution of the adsorbate (observable e.g.

in light scattering

54

).

Whether such large scale heterogeneities can also

explain the sharp contraction peak remains speculative. In any case it is possible that as a result of the reduction of the external pressure the mechanical stability limit of the adsorbate is exceeded and cavitation bubbles are formed in the bigger pore section.

51

The appearance

of small vapor bubbles near the center of the bigger pore sections involves the formation of a thick surface layer on the walls of these blocked pore sections. According to Eqs. 3 and 5 such a conguration leads also to a low value of the solvation pressure because of the related low Laplace pressure

−γlv / (rP − t).

cavitation at the pore walls (with ink-bottle pores)

56

55,56

On the other hand also the occurance of heterogeneous

that was apparantly observed for argon in porous silicon

corresponds to a conguration with no surface layer on the walls

of some pore sections and would result in a strong contraction peak.

Conclusions We have expanded the model by Gor and Neimark describing sorption-induced deformation to account for the coexistence of capillary bridges and adsorbed layers. Our model is in the strict sense valid for porous samples with a constant pore radius. Consequently, for porous samples with a sharp monomodal pore size distribution it is possible to determine the geometrical arrangement during sorption of the adsorbate comparing the measured deformation and the theoretical behavior. But also the measured deformation behavior of porous Vycor

R

glass during sorption of argon can be explained at least qualitatively. The comparison between theory and experiment shows that during adsorption 2 to 4 surface layers are stable. The analysis of the sharp contraction peak during the draining of the pores suggests that

22


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Langmuir

only a thin surface layer (at maximum one monolayer) is present on the pore walls as long as there are capillary bridges. Such a conguration would not correspond to a stable equilibrium state. Pore blocking eects or caviation during the draining of the polydisperse network structure of Vycor

R

might also cause the contraction/re-expansion peak. Thus, deformation

measurements can also help to achieve information on the draining mechanism of porous samples. The ultrasonic measurements show that the elastic properties change signicantly during lling (5 − 10%, see Figs. 5c and 6b+d). But even though the elastic properties are directly related to the deformation behavior, the observed change of the elastic properties during sorption seems not to be the major reason for the strong contraction peak during desorption. The fact that the sharp contraction peak during desorption has not been observed for other systems raises the question for the reason of this dierence, e.g. during the desorption of water from the pores of porous Vycor

R

no sharp contraction was observed.

19

The analysis and

discussion suggests that (long-lasting) metastable or unequilibrated states can inuence the observed deformation behavior signicantly. Also other interactions between adsorbate and the porous sample (cp. e.g. Ref.

58

), dierent structures of the pore walls and dierent pore

size distributions are related to changes in the desorption mechanism as well as the stability of adsorbed surface layers and might thus account for dierent deformation behaviors. The analysis method presented in this article can together with further measurements help to resolve the correlations between the conguration of the adsorbate, the interaction between adsorbate and porous sample, and the deformation behavior.

References (1) Gor, G. Y.; Neimark, A. V. Adsorption-Induced Deformation of Mesoporous Solids. Langmuir 2010, 26, 13021-13027.

(2) Yang, K.; Lu, X.; Lin, Y.; Neimark, A. V. Deformation of Coal Induced by Methane Adsorption at Geological Conditions. Energy Fuels 2010, 24, 5955-5964.

23


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(3) White, C. M.; Smith, D. H.; Jones, K. L.; Goodman, A. L.; Jikich, S. A.; LaCount, R. B.; DuBose, S. B.; Ozdemir, E.; Morsi, B. I.; Schroeder, K. T. Sequestration of Carbon Dioxide in Coal with Enhanced Coalbed Methane Recovery A Review. Energy Fuels 2005, 19, 659-724.

(4) Brochard, L.; Vandamme, M.; Pellenq, R. J.-M.; Fen-Chong, T. Adsorption-Induced Deformation of Microporous Materials: Coal Swelling Induced by CO 2 CH4 Competitive Adsorption. Langmuir 2012, 28, 2659-2670.

(5) Karacan, C. O. Heterogeneous Sorption and Swelling in a Conned and Stressed Coal during CO2 Injection. Energy Fuels 2003, 17, 1595-1608.

(6) Makal, T. A.; Li, J.-R.; Lu, W.; Zhou, H.-C. Methane storage in advanced porous materials. Chem. Soc. Rev. 2012, 41, 7761-7779.

(7) Günther, G.; Prass, J.; Paris, O.; Schoen, M. Novel Insights into Nanopore Deformation Caused by Capillary Condensation. Phys. Rev. Lett. 2008, 101, 086104.

(8) Gor, G. Y.; Neimark, A. V. Adsorption-Induced Deformation of Mesoporous Solids: Macroscopic Approach and Density Functional Theory. Langmuir 2011, 27, 6926-6931.

(9) Gor, G. Y.; Paris, O.; Prass, J.; Russo, P. A.; Ribeiro Carrott, M. M. L.; Neimark, A. V. Adsorption of n-Pentane on Mesoporous Silica and Adsorbent Deformation. Langmuir 2013, 29, 8601-8608.

(10) Erko, M.; Wallacher, D.; Paris, O. Deformation mechanism of nanoporous materials upon water freezing and melting. Appl. Phys. Lett. 2012, 101, 181905.

(11) Erko, M.; Wallacher, D.; Findenegg, G. H.; Paris, O. Repeated sorption of water in SBA-15 investigated by means of in situ small-angle x-ray scattering. J. Phys.: Condens. Matter 2012, 24, 284112.

24


Page 24 of 31

Page 25 of 31

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(12) Zickler, G. A.; Jähnert, S.; Funari, S. S.; Findenegg, G. H.; Paris, O. Pore lattice deformation in ordered mesoporous silica studied by in situ small-angle X-ray diraction. J. Appl. Cryst. 2007, 40, s522-s526.

(13) Vandamme, M.; Brochard, L.; Lecampion, B.; Coussy, O. Adsorption and strain: The CO2 -induced swelling of coal. J. Mech. Phys. Solids 2010, 58, 1489-1505.

(14) Kowalczyk, P.; Furmaniak, S.; Gauden, P. A.; Terzyk, A. P. Methane-Induced Deformation of Porous Carbons: From Normal to High-Pressure Operating Conditions. J. Phys. Chem. C 2012, 116, 1740-1747.

(15) Balzer, C.; Wildhage, T.; Braxmeier, S.; Reichenauer, G.; Olivier, J. P. Deformation of Porous Carbons upon Adsorption. Langmuir 2011, 27, 2553-2560.

(16) Schoen, M.; Paris, O.; Günther, G.; Müter, D.; Prass, J.; Fratzl, P. Pore-lattice deformation on ordered mesoporous matrices: experimental studies and theoretical analysis. Phys. Chem. Chem. Phys. 2010, 12, 11267-11269.

(17) Herman, T.; Day, J.; Beamish, J. Deformation of silica aerogel during uid adsorption. Phys. Rev. B 2006, 73, 094127.

(18) Dourdain, S.; Britton, D. T.; Reichert, H.; Gibaud, A. Determination of the elastic modulus of mesoporous silica thin lms by x-ray reectivity via capillary condensation of water. Appl. Phys. Lett. 2008, 93, 183108.

(19) Amberg, C. H.; McIntosh, R. A Study of Adsorption Hysteresis by Means of Length Changes of a Rod of Porous Glass. Can. J. Chem. 1952, 30, 1012-1032.

(20) Grosman A.;

Ortega,

C. Inuence of elastic deformation of porous materials in

adsorption-desorption process: A thermodynamic approach. Phys. Rev. B 2008, 78, 085433.

25


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(21) Dolino G.; Bellet, D.; Faivre C. Adsorption strains in porous silicon Phys. Rev. B 1996, 54, 17919-17929.

(22) Haines, R. S.; McIntosh, R. Length changes of activated carbon rods caused by adsorption of vapors. J. Chem. Phys. 1947, 15, 2838.

(23) Prass, J.; Müter, D.; Fratzl, P.; Paris, O. Capillary-driven deformation of ordered nanoporous silica. Appl. Phys. Lett. 2009, 95, 083121.

(24) Yates, D. J. C. Volume changes in porous glass produced by the physical adsorption of gases. In Proceedings of the International Congress on Catalysis. Philadelphia, Pennsylvania, 1956; Farkas, A., Ed.; Advances in Catalysis and Related Subjects; Academic

Press: New York, N. Y., 1957; pp 481-487.

(25) Schappert K.; Pelster, R. Inuence of the Laplace pressure on the elasticity of argon in nanopores. EPL 2014, 105, 56001.

(26) Data sheet for porous Vycor 7930, Corning, Inc.

(27) Schüth, F.; Sing, K. S. W.; Weitkamp, J., Eds. Handbook of Porous Solids; Wiley-VCH: Weinheim, 2002; Vol. 3

(28) Gregg, S.; Sing, K. Adsorption, Surface Area, and Porosity; Academic Press: London, 1982.

(29) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids; Academic Press: London, 1999.

(30) Lowell, S.; Shields, J. E.; Thomas, M. A.; Thommes, M. Characterization of Porous Solids and Powders: Surface Area, Pore Size and Density; Kluwer Academic Publishers:

Dordrecht, 2004.

26


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(31) Levitz, P.; Ehret, G.; Sinha, S. K.; Drake, J. M. Porous vycor glass: The microstructure as probed by electron microscopy, direct energy transfer, smallangle scattering, and molecular adsorption. J. Chem. Phys. 1991, 95, 6151-6161.

(32) Before a lling step the gas distribution system (with the volume ature

TGDS )

is lled with a certain pressure of gas,

is closed). In addition, there is some vapor (pressure volume

VSC

at the temperature

piGDS

VGDS

at the temper-

(the valve to the sample cell

0

pi−1 SC )

in the sample cell (with the

TSC ) and some adsorbate ∆ni−1 in the pores of the sam-

ple (if it is not the rst lling step). Thus, for the total molar amount of the molecules in the experimental setup before the opening of the valve the ideal gas equation yields:

0

ntotal

with the gas constant 0

0

piGDS = piSC

R.

i−1 VSC piGDS VGDS pSC + + ∆ni−1 , = RTGDS RTSC

After the opening of the valve a new equilibrium pressure

∆ni

is reached and some additional adsorbate

has formed:

0

0

ntotal

(8)

piGDS VGDS piSC VSC + + ∆ni−1 + ∆ni = RTGDS RTSC

Consequently, the amount of adsorbate,

(9)

∆ni , added in one lling step can be calculated

according to

∆ni =

VGDS i VSC i−10 0 0 pGDS − piGDS + pSC − piSC . RTGDS RTSC

For the consideration of the molecules in the capillary (with the volume temperature

TC )

the additional term

0

0

i−1 VC / (RTC ) pSC − piSC

(10)

VC

at the

has to be added on the

right hand side of equation 10.

(33) Valiullin, R.; Naumov, S.; Galvosas, P.; Kärger, J.; Woo, H.-J.; Porcheron, F.; Monson, P. A. Exploration of molecular dynamics during transient sorption of uids in

27


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Page 28 of 31

mesoporous materials. Nature 2006, 443, 965-968.

(34) Page, J. H.; Liu, J.; Abeles, B.; Herbolzheimer, E.; Deckman, H. W.; Weitz, D. A. Adsorption and desorption of a wetting uid in Vycor studied by acoustic and optical techniques. Phys. Rev. E 1995, 52, 2763-2777.

(35) Schappert K.; Pelster, R. Elastic properties of liquid and solid argon in nanopores. J. Phys.: Condens. Matter 2013, 25, 415302.

(36) A similar idea of introducing an empirical coecient related to the fraction of pore sections lled with capillary bridges was used in Refs..

15,37

(37) Reichenauer, G.; Scherer, G. W. Nitrogen sorption in aerogels. J. Non-Cryst. Solids 2001, 285, 167-174.

(38) Schappert K.; Pelster, R. Elastic properties and freezing of argon conned in mesoporous glass. Phys. Rev. B 2008, 78, 174108.

(39) Page, J. H.; Liu, J.; Abeles, B.; Deckman, H. W.; Weitz, D. A. Pore-Space Correlations in Capillary Condensation in Vycor. Phys. Rev. Lett. 1993, 71, 1216-1219.

(40) Schappert K.; Pelster, R. Freezing behavior of argon layers conned in mesopores. Phys. Rev. B 2011, 83, 184110.

(41) Schappert K.; Pelster, R. Continuous Freezing of Argon in Completely Filled Mesopores. Phys. Rev. Lett. 2013, 110, 135701.

(42) For the calculation of the solvation pressure at a specic external pressure the knowledge of the relation between the lm thickness

t

and the external pressure

because Eq. 3 gives the solvation pressure as a function of the thickness lm. For a pore with a constant pore diameter of

p

is necessary,

t of the adsorbed

rP = 4.0 nm (cp. Fig. 7) the Derjaguin

28


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Langmuir

equation for the chemical potential

µ

was used:

1

µ = RT ln(p/p0 ) = − Π(t) +

With the aid of Eq. 11 also the external pressure

γlv rP − t

Vm

p at which the pores ll abruptly with

capillary bridges can be determined (maximum of the chemical potential).

(43) The following values for bulk argon at vation pressure: (extrapolated).

(11)

1

T = 86 K were used for the calculation of the sol-

γlv = 12.9 mN/m (interpolated) 59

and

Vm = mM /ρads = 28.5 cm3 /mol

60

(44) Cole, M. W.; Saam, W. F. Excitation Spectrum and Thermodynamic Properties of Liquid Films in Cylindrical Pores. Phys. Rev. Lett. 1974, 32, 985-988.

(45) Saam, W. F.; Cole, M. W. Excitations and thermodynamics for liquid-helium lms. Phys. Rev. B 1975, 11, 1086-1105.

(46) Woo, H.-J.; Sarkisov, L.; Monson, P. A. Mean-Field Theory of Fluid Adsorption in a Porous Glass. Langmuir 2001, 17, 7472-7475.

(47) 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, 055701.

(48) Neimark, A. V.; Vishnyakov, A. Gauge cell method for simulation studies of phase transitions in conned systems. Phys. Rev. E 2000, 62, 4611-4622.

(49) Reichenbach, C.; Kalies, G.; Enke, D.; Klank, D. Cavitation and Pore Blocking in Nanoporous Glasses. Langmuir 2011, 27, 10699-10704.

(50) Klomkliang, N.; Do, D. D.; Nicholson, D. On the hysteresis loop and equilibrium transition in slit-shaped ink-bottle pores. Adsorption 2013, 19, 1273-1290.

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(51) Sarkisov, L.; Monson, P. A. Modeling of Adsorption and Desorption in Pores of Simple Geometry Using Molecular Dynamics Langmuir 2001, 17, 7600-7604.

(52) Nguyen, P. T. M.; Fan, C.; Do, D. D.; Nicholson, D. On the Cavitation-Like Pore Blocking in Ink-Bottle Pore: Evolution of Hysteresis Loop with Neck Size. J. Phys. Chem. C 2013, 117, 5475-5484.

1 (53) Cohan, L. H. Hysteresis and the Capillary Theory of Adsorption of Vapors . J. Am. Chem. Soc. 1944, 66, 98-105.

(54) Wallacher, D.; Soprunyuk, V. P.; Kityk, A. V.; Huber, P.; Knorr, K. Capillary sublimation of Ar in mesoporous glass. Phys. Rev. B 2005, 71, 052101.

(55) Rasmussen, C. J.; Gor, G. Y.; Neimark, A. V. Monte Carlo Simulation of Cavitation in Pores with Nonwetting Defects. Langmuir 2012, 28, 4702-4711.

(56) Grosman, A.; Ortega, C. Cavitation in Metastable Fluids Conned to Linear Mesopores. Langmuir 2011, 27, 23642374.

(57) If we assume only one adsorbed surface layer for the calculation of the pore radius (cp. discussion about the layer thickness and the introduction) we obtain an average pore radius of

≈ 3.3 nm or ≈ 3.5 nm. The presented results and calculations (see e.g. Figs. 9

and 10) are not notably inuenced by such a change of the pore radius.

(58) Schappert K.;

Pelster, R. Strongly enhanced elastic modulus of solid nitrogen in

nanopores. Phys. Rev. B 2013, 88, 245443.

(59) Stanseld, D. The Surface Tension of Liquid Argon and Nitrogen. Proc. Phys. Soc. 1958, 72, 854-866.

(60) van Itterbeek, A.; Verbeke, O. Density of liquid nitrogen and argon as a function of pressure and temperature. Physica 1960, 26, 931-938.

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Unexpected Sorption-Induced Deformation of Nanoporous Glass

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