Deformation mechanism of nanoporous materials upon water freezing and meltingMaxim Erko, Dirk Wallacher, and Oskar Paris Citation: Applied Physics Letters 101, 181905 (2012); doi: 10.1063/1.4764536 View online: http://dx.doi.org/10.1063/1.4764536 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/101/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase diagram of supercooled water confined to hydrophilic nanopores J. Chem. Phys. 137, 044509 (2012); 10.1063/1.4737907 Freezing of mixtures confined in silica nanopores: Experiment and molecular simulation J. Chem. Phys. 133, 084701 (2010); 10.1063/1.3464279 Capillarity-driven deformation of ordered nanoporous silica Appl. Phys. Lett. 95, 083121 (2009); 10.1063/1.3213564 Origin of the enthalpy features of water in 1.8 nm pores of MCM-41 and the large C p increase at 210 K J. Chem. Phys. 130, 124518 (2009); 10.1063/1.3103950 Dynamics of ultrathin metal films on amorphous substrates under fast thermal processing J. Appl. Phys. 102, 104308 (2007); 10.1063/1.2812560

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Deformation mechanism of nanoporous materials upon water freezingand melting

Maxim Erko,1,2 Dirk Wallacher,3 and Oskar Paris1,a)

1Institute of Physics, Montanuniversitaet Leoben, Franz-Josef-Strasse 18, 8700 Leoben, Austria2Department of Biomaterials, Max-Planck-Institute of Colloids and Interfaces, Research Campus Golm,14424 Potsdam, Germany3Helmholtz Zentrum Berlin f€ur Materialien und Energie, Hahn Meitner Platz 1, 14109 Berlin, Germany

(Received 7 September 2012; accepted 15 October 2012; published online 1 November 2012)

Temperature-induced non-monotonous reversible deformation of water-filled nanoporous silica

materials is investigated experimentally using in-situ small-angle x-ray scattering. The influence of

freezing and melting in the nanopores on this deformation is treated quantitatively by introducing a

simple model based on the Gibbs-Thomson equation and a generalized Laplace-pressure. The

physical origin of the melting/freezing induced pore lattice deformation is found to be exactly

the same as for capillary condensation/evaporation, namely the curved phase boundary due to the

preferred wetting of the pore walls by the liquid phase. As a practical implication, elastic properties

of the nanoporous framework can be determined from the temperature-deformation curves. VC 2012American Institute of Physics. [http://dx.doi.org/10.1063/1.4764536]

The phase behavior of fluids and condensed matter –

particularly that of water – in nanometer confinement is a

fascinating interdisciplinary topic, which attracts scientists

from fields as diverse as geology, biology, chemistry,

physics, and nanotechnology.1 The gas-liquid and the liquid-

solid transitions in nanopores are well known to be shifted

with respect to their bulk condensation pressure and their

bulk freezing temperature, respectively. The basic physics of

these phenomena is satisfactorily described by classical ther-

modynamics, i.e., by the Kelvin equation for capillary con-

densation and by the Gibbs-Thomson equation for the

melting point suppression.2 Much work has been devoted to

go beyond these simple pictures by sophisticated experi-

ments and by atomistic modeling and simulation attempts.3

However, still little recognized is the fact that the confining

porous framework is never infinitely stiff and will, therefore,

deform as a consequence of the interaction of the guest phase

with the solid pore walls. Sorption induced deformation of

porous solids is known already since many decades4,5 and

has recently attracted renewed interest, one goal being the

design of sensing and actuating materials.6 Experimentally,

such deformation has mainly been studied as a function of

gas pressure during adsorption and capillary condensation of

fluids in various nanoporous systems,7,8 and theoretical work

as well as simulations have been undertaken to describe the

observed non-monotonous deformation behavior with

pressure.9,10

Non-monotonous deformation of nanoporous materials

has also been observed for condensed matter transitions as a

function of temperature, i.e., as a consequence of freezing

and melting.11–14 It has been pointed out already by Everett12

and later on by Faivre et al.14 that the thermal expansion of

the bulk liquid and solid cannot explain the experimental

facts, as organic fluids with “normal” and water with

“anomalous” expansion showed a very similar deformation

behavior. As yet, in contrast to the well-established picture

for sorption strains due to the gas-liquid transition in nano-

pores, a coherent explanation for the pore wall deformation

due to a liquid-solid transition in confinement is still missing.

Based on experimental data from in-situ small-angle x-ray

scattering (SAXS) on water-filled cylindrical nanopores of

uniform diameter (MCM-41), this Letter provides a quantita-

tive explanation for this phenomenon. We highlight the strik-

ing similarity between the deformation as a consequence of

capillary condensation/evaporation and the melting/freezing

transition. In fact, we demonstrate that these two phenomena

rely on exactly the same physical mechanism, namely the

change of the curvature of a liquid meniscus in contact with

the gas phase or the solid phase, respectively.

The experimental details of in-situ SAXS measurements

at constant temperature or constant pore filling of ordered

nanoporous materials with cylindrical pores on a 2D hexago-

nal lattice were already reported elsewhere.7,10,15,16 MCM-

41 samples with different pore diameter ranging from 4.4 nm

down to 2.0 nm diameter were investigated in the tempera-

ture regime between 100 K and 280 K. The porous samples

were filled with water from the gas phase at 280 K to the

desired filling fraction within a volumetric adsorption sys-

tem. More experimental details are given in Ref. 16. The de-

formation of the porous systems was quantified by the shift

of the Bragg reflections originating from the pore lattice.

The pore lattice strain is ePL(T)¼ (q0 / q(T))� 1, where q0 is

the scattering vector length of the reference Bragg peak posi-

tion of the water filled sample at 280 K, and q(T) is the corre-

sponding peak position at temperature T. We note that by

this definition, the strain is arbitrarily set to zero at the begin-

ning of the first cooling cycle starting at 280 K.

Strain versus temperature curves for a full cooling and

heating cycle are exemplarily shown in Fig. 1 for a sample

with 4.4 nm pore diameter at a nominal water filling fraction

f¼ 0.99. In contrast to the empty (evacuated) sample also

shown in Fig. 1, the pore lattice deformation of the water-

filled sample is not monotonous with temperature within the

a)Author to whom corresponding should be addressed. Electronic mail:

oskar.paris@unileoben.ac.at.

0003-6951/2012/101(18)/181905/4/$30.00 VC 2012 American Institute of Physics101, 181905-1

APPLIED PHYSICS LETTERS 101, 181905 (2012)

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studied region. Four roughly linear regimes with different

magnitude and sign of the slope of the ePL(T) curve can be

identified (see inset in Fig. 1). Starting with the heating cycle

at low temperature, ePL increases with a slope about a factor

3 higher as compared to the empty sample (region I). This

increase, extending over more than 100 K, is continued by an

abrupt decrease of ePL within a narrow temperature interval

of less than 10 K (region II). It follows a linear increase

(region III) and a leveling off (region IV). Upon cooling, a

qualitatively similar behavior is found, but a broad hysteresis

is present in region I, and in region II-IV, the two curves are

also shifted with respect to the temperature axis. The strain

minima separating regions II and III upon cooling and heat-

ing correspond exactly to the freezing (Tf) and melting (Tm)

temperatures of water confined in the 4.4 nm pores, as deter-

mined independently by differential scanning calorimetry

(DSC)17 and Raman scattering.18

This general behavior is valid also for samples with

smaller pores (3.9 nm and 3.4 nm diameter). In particular, the

dip at the border between the regions II and III shifts system-

atically towards lower temperatures with decreasing pore size

and coincides perfectly with the corresponding water freezing

and melting temperatures in confinement.17,18 Thus, the sharp

step in region II clearly indicates freezing/melting of the con-

fined pore water. The shallow increase of ePL in region I can,

therefore, simply be attributed to the (normal) thermal expan-

sion of ice confined within the pores. Since the thermal expan-

sion coefficient of ice is much larger than the one of silica, the

larger slope for the water filled samples as compared to the

empty silica sample in region I is reasonable. However, a

quantitative treatment is difficult since it is known that about

two monolayers of non-freezable water cover the pore walls

even at very low temperatures. A dynamical transition of this

liquid-like layer around 180 K has been proposed recently,16,19

which may be connected to the hysteresis seen in region I. We

further note that the basic features sketched in the inset in Fig. 1

are recognized also for samples with pore sizes below 3 nm.

Since for such small pores, no first order transition of water

in confinement takes place,16–18 and because the features are

strongly broadened with no clearly defined crossover

between the different regions, we do not include these data

in the subsequent analysis.

The pronounced contraction/expansion of the pore lattice

upon heating/cooling in region II obviously suggests assigning

this effect to the anomalous volume change between the liquid

and solid phases of the core water in the pores. However, sim-

ilar observations with the same sign of the deformation within

this region were reported from measurements with organic flu-

ids showing normal volume change upon freezing and melt-

ing.11,14 Moreover, after melting of the pore water, its volume

change with increasing temperature in region III should be

negative, and thus the progressively expansive deformation of

the pore lattice in this region is at least counterintuitive. Most

noteworthy, the strain curves within region III are all linear

with comparable slopes upon cooling and heating. This slope

decreases systematically with decreasing pore size. Before we

can introduce an explanation of the deformation behavior in

regions II and III, however we have first to answer a question

related to region IV.

The crossover from region III to IV upon heating coin-

cides exactly with the bulk water melting temperature of

273 K irrespective of the pore size, while this point is shifted

to lower temperatures (250–260 K) for the cooling cycle.

This indicates the freezing and melting of bulk ice at these

points upon cooling and heating, respectively, in the case of

freezing including a certain degree of supercooling. Since

the samples were filled from the gas phase to a nominal fill-

ing fraction of f¼ 0.99, there should, however, be no excess

water available outside the pores allowing bulk ice forma-

tion. Nevertheless, since the pores are open, they can take up

water molecules from the gas phase in the surrounding con-

tainer upon cooling. Moreover, since the water in the pores

expands below 277 K with decreasing temperature, this leads

also to an increase of the pore filling fraction. A rough esti-

mate together with the uncertainty related to the determina-

tion of the nominal filling fraction shows that condensation

of water outside the pores at the surface of the micrometer

sized MCM-41 powder grains is possible. In order to check

this experimentally, additional measurements with different

nominal pore filling fractions were performed for one of the

samples. Fig. 2 shows the strain curves obtained for the heat-

ing cycle at nominally f¼ 0.1 and f¼ 0.6 (partially filled),

f¼ 0.99 (filled), and f> 1 (overfilled). The filled and the

overfilled samples show almost perfect agreement in the

regions II–IV. Only in region I, the slope is different, which

may be related to the additional external constraint that in

the overfilled sample the whole grain is fully embedded into

bulk ice at low T. For a filling fraction of f¼ 0.6, there is no

more kink at 273 K, indicating that there is no more bulk ice

present, as expected. At this partially filled state, the meso-

pores are contracted already at high temperatures due to the

presence of concave liquid-vapor menisci.7 The slope of the

straight line in region III is similar to that of the empty

sample, the deviation being tentatively attributed to a

temperature-induced change of the interfacial energy

FIG. 1. Pore lattice strain versus temperature for the sample with 4.4 nm

pore diameter. The black circles are from the empty sample (filling fraction

f¼ 0). The blue (triangles down) and red (triangles up) curves stem from the

water filled sample (f¼ 0.99) for the cooling and heating cycles, respec-

tively. Freezing (Tf) and melting (Tm) temperatures of the confined pore

water are indicated, and the vertical dashed line is at 273 K. The inset

sketches the four temperature regions I–IV.

181905-2 Erko, Wallacher, and Paris Appl. Phys. Lett. 101, 181905 (2012)

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between the pore liquid and the surrounding vapor phase. This

deviation vanishes for filling fractions as low as f¼ 0.1 where

only a water film covers the pore walls.17 Here, the strain

curves show a roughly linear increase of the strain with tem-

perature for the whole region, the slope matching the one of

the empty sample. A deeper analysis on how temperature

affects the deformation behavior of partly filled samples goes,

however, beyond the scope of this Letter. The following dis-

cussion, therefore, focuses solely on water-filled nanopores in

contact with a bulk ice phase outside the pores.

We now return to the deformation behavior in regions II

and III. There is a remarkable similarity between the strain

as a function of temperature in water filled samples shown in

Figs. 1 and 2, and the “strain-isotherms” obtained during

adsorption and condensation of fluids as a function of the

reduced vapor pressure p/p0 at constant temperature T (see,

e.g., Fig. 1 in Ref. 7 or Fig. 6 in Ref. 10). In Ref. 7, the situa-

tion was interpreted within a classical thermodynamic pic-

ture with the experimentally observed pore lattice strain ePL

assumed to be proportional to the Laplace pressure P due to

a hemispherical liquid-vapor interface in the cylindrical

pores.

P ¼ 2ccoshR¼ MPL ePL; (1)

c¼ cLV is the liquid-vapor specific interfacial energy (surface

tension) and h is the wetting angle of the liquid phase. The

proportionality constant was denoted “pore load modulus”

MPL, which takes into account the peculiar loading situation

of the porous framework by a negative pressure within each

single pore, as well as the fact that the Laplace pressure

might not directly correspond to the bulk stress within the

pore walls. Combining Eq. (1) with the Kelvin equation

lnðp=p0Þ ¼ P Vm=ðRGTÞ, a direct proportionality between

ePL and ln(p/p0) results

ePL ¼1

MPL

RGT

Vmlnðp=p0Þ; (2)

Vm is the molar volume of the liquid phase and RG and T the

gas constant and temperature, respectively. Equation (2)

allows determining MPL from experimental data without

detailed knowledge of the surface tension cLV, the wetting

angle h, or the meniscus curvature radius R. It is even not

necessary to assume any details of the liquid-vapor interface.

A pictorial explanation of the strain isotherm for a fluid fully

wetting the pore walls is sketched in Fig. 3. Region I is

explained by the so called “Bangham effect,”4 i.e., the

adsorption of a wetting fluid at the pore walls lowers the

interfacial energy and leads to an expansive deformation of

the porous material. Region II is characterized by the sponta-

neous filling of cylindrical pores with radius R, with the neg-

ative Laplace pressure inducing a contractive deformation of

the pore lattice. In region III, finally, the liquid-vapor inter-

face is pinned at the pore ends, where the change of the cur-

vature according to the Kevin equation leads to an expansion

of the pore lattice with pressure until p¼ p0. The finite width

of region II is due to a (though small) pore size distribution

of the used MCM-41 materials. It has been shown by Schoen

et al.10 that the functional analysis of such data is even not

restricted to the presence of a curved liquid-vapor interface,

as an expression equivalent to Eq. (2) can be derived from a

thermodynamic analysis without taking any interfacial ge-

ometry into account. Nonetheless, we further stick to the

very useful concept of the curved interface to ease the fol-

lowing discussion for liquid-solid interfaces.

A liquid that wets the pore wall material in equilibrium

with its solid phase will be thermodynamically favored in

confinement. This is the simple physical reason for the melt-

ing point suppression of liquids in confinement. Formally,

the concept of the Laplace pressure from Eq. (1) is equally

applicable to a hemispherical liquid-solid interface within a

cylindrical pore, with c¼ cLS being now the liquid-solid spe-

cific interfacial energy.2 Within classical nucleation theory,

this quantity is formally furthermore connected to the Gibbs-

Thomson equation DT ¼ T0 � T ¼ 2cLSVmT0=ðR HmÞ, T0

being the bulk melting temperature and T the melting

FIG. 2. Lattice strain versus temperature for the sample with 3.9 nm pore

diameter during heating for four different nominal pore filling fractions.

f¼ 0.1 (black/circle), f¼ 0.6 (green/ diamond), f¼ 0.99 (red/triangle up),

and f> 1 (magenta/ squares). The melting temperature of water within the

pores (233 K)17,18 and the bulk melting temperature of water (273 K) are

indicated by vertical lines.

FIG. 3. Schematic view of the analogy between the vapor-liquid (upper

part) and the solid-liquid (lower part) phase transition mechanism in cylin-

drical nanopores, leading to the observed deformation behavior of the pore

lattice in regions II and III (middle part). The silica walls (dark grey) are in

contact with the (wetting) liquid water phase (blue/light grey) and with the

water vapor (upper part) or the solid ice phase (lower part) in white.

181905-3 Erko, Wallacher, and Paris Appl. Phys. Lett. 101, 181905 (2012)

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temperature of a (hemi-) spherical solid particle with radius

R, and Hm is the molar melting enthalpy. Assuming again

that the deformation of the pore lattice upon melting is pro-

portional to the generalized Laplace pressure given in Eq.

(1) with c¼ cLS, we obtain

ePL ¼1

MPL

Hm

VmT0

DT: (3)

This equation is very similar in its structure to Eq. (2) and

predicts a linear increase of the pore lattice strain with tem-

perature as experimentally observed in region III of our data

(see Figs. 1 and 2). The geometric situation is again sketched

in Fig. 3, with the solid-liquid transition in confinement

being fully analogous to the vapor-liquid transition in

regions II and III. Upon melting, in region II, a liquid bridge

forms at Tm out of the solid phase and grows parallel to the

pore axis until it reaches the pore end. Here, the liquid is

now in contact with bulk ice outside the pore via a hemi-

spherical meniscus creating a Laplace pressure, which leads

to the abrupt pore lattice contraction. In region III, the curva-

ture of this meniscus increases with temperature according to

the Gibbs Thomson equation, leading to a linear increase of

the strain with temperature until T reaches T0¼ 273 K. But

obviously, the analogy sketched in Fig. 3 does not extend to

region I, which is described by the Bangham effect for

adsorption and by thermal expansion of pore ice for heating

(see discussion above). It does also not apply to region IV,

where the detailed behavior will be strongly dependent on

the outer shape of the pore mouths and the grains and the

amount of water outside the pores. We stress once more that

the picture sketched in Fig. 3 for the liquid-solid transition

breaks down for partially filled pores since then, the pore liq-

uid will not anymore be in equilibrium with the bulk solid

phase outside the pores.

Equation (3) allows deducing the pore load modulus

from the slope of the strain-temperature curve in region III

(see Fig. 1), again without any detailed knowledge of the

geometric and energetic properties of the interface between

the two phases being required. Table I lists the values of MPL

obtained from the average of the cooling and heating cycles.

We note that there is no systematic difference of MPL

between heating and cooling, suggesting the concept to be

valid for both, the melting and the freezing transition. We

conjecture that the details of the hysteresis and the differen-

ces in region II may be rationalized by a more rigorous ther-

modynamic treatment,20 which is out of the scope of this

Letter. Table I also lists the MPL values obtained for the

liquid-vapor transition of pentane in the same samples using

Eq. (2).7 There is a systematic difference, with the values

obtained in this work being about 15% larger. This differ-

ence is not completely understood, since the determination

of MPL with Eqs. (2) and (3) does not require the knowledge

of any specific interfacial energy of the involved phases, but

only well-known fluid quantities. It is interesting to note that

for capillary evaporation, the pore load modulus for the ma-

terial with 3.9 nm pores was also larger for water (14.7 GPa)

as compared to pentane (13.9 GPa). Therefore, we attribute

this difference to the more complex interaction of water with

the pore walls, as chemical interactions would make the

application of the Kelvin- and the Gibbs-Thomson equations

more questionable.

We are grateful to G. H. Findenegg for discussions and

providing the samples; to A. Hoell, S. Haas, and D. Tatchev

for help with the experiments at the SAXS beamline of the

BESSY II storage ring at HZB Berlin; and to the European

LIght Sources Activity (ELISA) for financial support.

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TABLE I. Pore load modulus obtained from the freezing/melting of con-

fined water. The influence of the empty sample was corrected, and the values

used to calculate MPL with Eq. (3) were Hm¼ 6.01 kJ/mol, Vm¼ 1.8 10�5

m3/mol, and T0¼ 273 K.17 The modulus values are compared with those

obtained from the evaporation of n-pentane (C5H12) using Eq. (2) within the

same samples.7

1 Pore diameter nm MPL (this work) GPa MPLa GPa

2 3.4 18.9 6 1.2 15.8

3 3.9 16.5 6 0.3 13.9

4 4.4 12.8 6 1.5 11.2

aReference 7.

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