Solvent Cavitation during Ambient Pressure Drying of Silica
Aerogels
Julien Gonthier,*Ernesto Scoppola, Tilman Rilling, Aleksander Gurlo, Peter Fratzl,
and Wolfgang Wagermaier*
Cite This: Langmuir 2024, 40, 12925−12938
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ABSTRACT: Ambient-pressure drying of silica gels stands out as
an economical and accessible process for producing monolithic
silica aerogels. Gels experience significant deformations during
drying due to the capillary pressure generated at the liquid−vapor
interface in submicron pores. Proper control of the gel properties
and the drying rate is essential to enable reversible drying
shrinkage without mechanical failure. Recent in operando
microcomputed X-ray tomography (μCT) imaging revealed the
kinetics of the phase composition during drying and spring-back.
However, to fully explain the underlying mechanisms, spatial
resolution is required. Here we show evidence of evaporation by
hexane cavitation during the ambient-pressure drying of silylated
silica gels by spatially resolved quantitative analysis of μCT data supported by wide-angle X-ray scattering measurements. Cavitation
consists of the rupture of the pore liquid put under tension by capillary pressure, creating vapor bubbles within the gels. We found
the presence of a homogeneously distributed vapor-air phase in the gels well ahead of the maximum shrinkage. The onset of this
vapor/air phase corresponded to a pore volume shrinkage of ca. 50 vol % that was attributed to a critical stiffening of the silica
skeleton enabling cavitation. Our results provide new aspects of the relation between the shape changes of silica gels during drying
and the evaporation mechanisms. We conclude that stress release by cavitation may be at the origin of the resistance of the silica
skeleton to drying stresses. This opens the path toward producing larger monolithic silica aerogels by fine-tuning the drying
conditions to exploit cavitation.
■INTRODUCTION
Aerogels are porous materials with high specific surface area
and submicrometer pore size.
1,2
They consist of a solid matrix
of particles or fibers filled by an open network of air-filled
pores making up 80−99% of the total volume, thus the name
aerogel.
3
Silica-based aerogels display among the lowest
thermal conductivities observed in solids,
4,5
making them
competitive in thermal insulation.
6,7
Aerogels are obtained by
replacing the pore liquid of a gel with air while conserving the
native matrix and pore structure, which is achieved by drying.
Ambient-pressure drying (also called evaporative drying)
stands out as a safe and economically attractive process to
produce aerogels granulates, composites, or monoliths
compared to more energy-intensive techniques like super-
critical drying.
8,9
Understanding, predicting, and potentially
tailoring the evaporation mechanisms in gels is crucial as they
impact the structure and performance of the aerogels. Previous
work revealed the kinetics of the evaporative drying process in
terms of global phase composition, giving insights into the
conditions of emergence and dynamic of the aerogel spring-
back.
10
To elucidate the actual evaporation mechanisms and
differentiate between evaporation by meniscus recession,
drying shrinkage, and cavitation, we modeled the spatial
distribution of the pore liquid and gas in the gels throughout
drying.
In general, evaporation of a liquid confined in a porous
media can occur by three mechanisms: evaporation by
recession of the liquid−vapor interface, drying shrinkage, and
cavitation (Figure 1). Drying starts with the formation of a
meniscus at the outer surface of the pores. This liquid−vapor
interface is subject to capillary pressure and the tension of the
liquid confined to a cylindrical pore of radius rccan be
calculated with the Young−Laplace equation:
11,12
= =p p p
r
2 cos
0 l
LV
c
(1)
Received: February 12, 2024
Revised: May 13, 2024
Accepted: May 21, 2024
Published: June 12, 2024
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Langmuir 2024, 40, 12925−12938
This article is licensed under CC-BY 4.0
where Δpis the capillary pressure, p0is the vapor pressure, plis
the pressure in the liquid, γLV is the surface tension of the
liquid−vapor interface, and θis the contact angle of the
meniscus. For wetting fluids (θ< 90°), plcan be negative: the
liquid is under tension.
13,14
Let us consider a liquid fully
wetting the solid (θ= 0, which is true for most solvent-gel
systems
15
). Evaporation by the recession of the meniscus will
occur, provided that the tension in the liquid remains under
the elastic limit of the solid matrix. This is typically the case for
porous materials with a large pore size or a stiff matrix. With
water as the pore liquid (γwater≈72 mN m−1) and a pore size of
1μm, the capillary pressure given by eq 1 would be around
0.15 MPa, which is about the yield strength of polyurethane
foams
16
(found in, e.g., kitchen sponges). The meniscus
recedes in the larger pores first, resulting in a heterogeneous
drying front inside the porous medium where evaporation is
limited by the diffusion of the vapor phase.
17,18
Under certain
conditions, evaporation by the recession of the meniscus may
also proceed by a sudden conversion of liquid to vapor referred
to as adiabatic burst events.
19
Evaporation by drying shrinkage takes place in porous
materials with a compliant matrix and a smaller pore size,
eventually preceded by an initial stage of evaporation by
meniscus recession. As the meniscus reaches smaller pores, the
tension in the liquid increases and can overcome the elastic
limit of the solid matrix, which then contracts onto the
liquid.
20
The flow of liquid resulting from the pore volume
shrinkage sustains the evaporation at the meniscus through
poromechanical coupling.
21
The capillary pressure can be as
high as 150 MPa for a pore radius of 1 nm (with eq 1,
considering water), resulting in a significant compressive stress
on the solid matrix.
Evaporation by cavitation may take place in porous materials
with a stiff matrix and “ink-bottle″-shaped pores, that is, large
pores constricted by smaller pore necks.
22
Unlike evaporation
by drying shrinkage, the tension in the liquid is not
compensated by a contraction of the solid matrix and keeps
rising, making the liquid metastable and susceptible to
cavitation. Cavitation occurs by the nucleation of vapor
bubbles in larger pores ahead of the pore constriction, where
the effects of confinement by the matrix are less strong.
23−25
The evaporation proceeds at the meniscus and is sustained by
the liquid flow toward the pore constriction at the expense of
the formation and growth of the bubbles. The onset of
cavitation depends on the state of the liquid: its saturation
vapor pressure, surface tension, and temperature;
26−29
and on
the properties of the porous media: the pore size
distribution,
22,26,27,30
the stiffness of the solid matrix,
23,25
and
the presence of defects.
31
Evaporative drying of gels is commonly described as a
succession of evaporation by drying shrinkage and meniscus
recession.
15
The drying stress is reduced by replacing the pore
liquid of the gels before drying with a liquid with a low surface
tension such as n-hexane.
32
Because the matrix of the gels is
initially rather compliant, evaporative drying first proceeds by
drying shrinkage, and the menisci remain mostly located on the
outer surface of the gel. The decrease in the pore volume
causes a progressive stiffening of the gel. As the tension in the
liquid increases, it reaches a maximum once the radius of
curvature of the meniscus becomes equal to the radius of the
smallest pores. At this point, the liquid tension cannot
overcome the stiffening of the solid matrix, marking the end
of drying shrinkage and the beginning of evaporation by
meniscus recession. This threshold is known as the maximum
shrinkage or critical point of drying where the gel volume can
reach about 20% of its original volume.
33,34
As the meniscus
recedes into the pores, the compressive stress on the solid
matrix related to capillary forces is released. In silica gels
modified with a silylating agent such as trimethylchlorosilane
(TMCS), this coincides with a re-expansion of the solid matrix,
enabling recovery of the drying shrinkage known as the spring-
back effect (SBE) and resulting in hydrophobic aerogels.
1
The
SBE is a key feature in the production of aerogels by
evaporative drying and its origin has not been investigated until
recently.
10,35−38
Nevertheless, it is not yet understood how
monolithic silica aerogels can be produced by evaporative
drying considering the extreme stress exerted on the silica solid
matrix by drying shrinkage and spring-back, which therefore
challenges the current models of evaporative drying.
In addition to the two known evaporation mechanisms, it is
theoretically possible that cavitation also plays a central role
during the drying of gels at ambient pressure as suggested by
Scherer & Smith.
39
Using classical nucleation theory (CNT),
they estimated that homogeneous nucleation of vapor bubbles
may happen before maximum shrinkage provided a small
enough pore size (1−2 nm). Cavitation events are particularly
intriguing in gels as they stabilize the tension in the liquid,
which in turn would reduce the stress on the silica matrix and
potentially prevent mechanical failure. A systematic literature
review of publications citing the Scherer & Smith paper
revealed the absence of experimental evidence of cavitation
events in the drying of gels to produce aerogels. To our
knowledge, there have been only three studies reporting visual
observations of an opaque phase growing in the core of gels
dried at ambient pressure that could possibly be associated
with cavitation bubbles.
40−42
The lack of investigation on
cavitation is presumably due to experimental limitations:
monitoring the evaporative drying of gels requires non-
destructive in operando methods at ambient pressure and
with enough resolution and/or contrast to resolve cavitation
events.
We have recently reported that monolithic silica aerogels
dried at ambient pressure contained up to 37 vol % of gas at
the maximum shrinkage using an X-ray microcomputed
tomography (μCT) quantitative imaging procedure.
10
This is
inconsistent with a drying model based on dual-evaporation
Figure 1. Illustration of the three evaporation mechanisms in a
simplified cylindrical pore with an ink-bottle-shaped geometry
confined by a solid matrix (in gray). (a) Evaporation by recession
of the meniscus. (b) Evaporation by drying shrinkage. (c)
Evaporation by homogeneous (center bubble) and heterogeneous
(edge bubble) cavitation. The black arrows depict the tension in the
liquid and the light blue arrows indicate the liquid flow. The gray
background represents the stiffness of the matrix, a denser background
stands for a higher stiffness. The wavy arrow on top illustrates the
evaporated liquid.
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mechanisms although the spatial distribution of gas was not
evaluated. This finding echoed with the theoretical study of
Scherer & Smith and motivated the current work with the aim
of evaluating cavitation as an additional evaporation mecha-
nism. μCT records changes in the absorption of an irradiated
specimen and allows the reconstruction of a 3D volume, in
which the contrast is proportional to the attenuation
coefficient of the specimen.
43
Quantitative imaging of the
reconstructed volumes correlates the temporal variations in the
attenuation coefficients of a specimen with changes in
composition by image processing and subsequent modeling.
Spatially resolved quantitative imaging can produce composi-
tion maps notably showing the distribution of vapor/air inside
of the gels and provide information on the evaporation
mechanisms. Besides absorption, changes in gels’ composition
can also be evaluated from the scattering of X-rays upon
irradiation of a gel. In the wide-angle X-ray scattering (WAXS)
region, the scattering signal arises from the molecular structure
of the pore liquid and solid silica skeleton.
35,44
Deconvolution
of these two signals and subsequent modeling allow us to
calculate the average phase composition of the gels within the
incident beam path.
In this work, previously published μCT data on the drying of
silica gels
10
were reanalyzed based on a spatially resolved μCT
quantitative imaging workflow to test for the hypothesis of
cavitation. While our precedent quantitative imaging workflow
only allowed us to calculate the average phase composition, the
procedure presented in this study generates composition maps
of pore liquid and vapor/air within silica gels during drying.
This computational approach consists in reducing and
interpolating 4D reconstructed volumes, along with a system-
atic evaluation of instrumental and computational artifacts.
Despite the resolution of the μCT scans (11 μm) being much
larger than the size of the cavitation bubbles as postulated by
Scherer & Smith (2 nm), the present approach was able to
capture the spreading of a vapor/air phase in the gels well
ahead of the maximum shrinkage. That gas phase appeared and
grew homogeneously across the sample, suggesting it was
created by cavitation of the solvent. In addition, we performed
WAXS measurements and CNT estimations, which consoli-
dated the results on the evaporation mechanisms. This study
reports indirect evidence of evaporation by cavitation in silica
gels by two methods, opening new aspects to the under-
standing and improvement of the evaporative drying process to
produce high-performance, monolithic aerogels.
■EXPERIMENTAL SECTION
Materials. Five silylated silica gels, previously analyzed to study
the kinetics of their average phase composition in ref 10, were reused
in this work with a new image processing approach. They were
synthesized by a two-step sol−gel process adapted from refs 35,45. A
silica sol was prepared from a tetraethyl orthosilicate (TEOS)
precursor and ethanol and was cast in cylindrical molds to produce
gels of 16 and 8 mm in height and diameter, respectively. The
resulting gels underwent a solvent exchange for n-hexane, followed by
a surface modification with trimethylchlorosilane (TMCS) in n-
hexane and a final solvent exchange for n-hexane. The five silylated
silica gels were labeled M1−M5.
In Operando μCT. The evaporative drying of gels M1−M5 was
monitored by taking a series of μCT scans as described in our recent
work.
10
Each scan was reconstructed and segmented using semi-
automatic procedures to generate masked slices of the specimen and
to calculate morphological information. This was done by transferring
a gel from its n-hexane storage solution to a tailored PEEK drying
chamber. The chamber has an open lid through which the vapors exit.
The chamber was placed in an EasyTom 160/150 CT system (RX
Solutions, Chavanod, France), and 141 μCT scans were acquired in
the step and shot mode without reference images at a voxel size of 11
μm to monitor the drying process in operando, which lasted between
14 and 16 h. The projections were reconstructed along the samples’
height using a cone-beam algorithm in the software XAct (RX
Solutions) to generate a set of slices (8-bit tif images). The
segmentation of the reconstructed volumes was carried out in the
software Dragonfly
46
using the Python console and a region of
interest (ROI) labeling the sample at each scan was created.
The ROIs were used to overwrite the gray values of all background
voxels in the slices by zero, generating masked slices that were
exported as 8-bit tif images. For a single sample, 141 series of 1000+
masked slices were created, corresponding to the segmented volume
of the gel over time. The total volume of the gel at each scan was
given by the total volume of the corresponding ROI: Vk, where the
index krefers to the scan number: 0 ≤k≤140 and
k
. The gels
volume was corrected as described in our previous work.
10
The drying
time was defined as the difference between the mean time of each
scan (taken as the average of the timestamps of the 64 projections)
and the time at which the lid of the drying chamber was opened. In
addition to the five silica gels, a reference liquid n-hexane sample was
scanned under the same in operando conditions. The masked images
were then processed using a quantitative imaging workflow consisting
of two parts: data reduction and modeling.
μCT Data Reduction. The motivation behind the reduction of the
reconstructed μCT data was to simplify the geometry of the gels
during drying, taking advantage of their cylindrical symmetry. Doing
so allowed us to work on a 2D or 3D data set instead of a 4D data set
(three spatial dimensions and one temporal dimension). It also
improved the presentation of the results that can be shown against
different axes of the cylinders independently. The time series of
masked images was reduced by integrating the data over one or more
axes of the samples. The raw gray values in the masked images were
corrected for the anode heel effect,
10,47
resulting in the gray values
gi,p,q,kwith i,p,qthe z,x,ycoordinates in the reconstructed volume,
respectively, and kthe time index.
i p q k,,,
. Formally, i=ik,p=
pk,and q=qkbecause the shape of the sample was changing over
time. Three reduction procedures were used to generate different
spatial and temporal representations of the reconstructed gray values
during drying (Figure 2):
a) Azimuthal integration (3D, Figure 2a): the masked slices were
integrated over the azimuth of the cylinder, resulting in 141 2D
gray value maps along the height and radius of the cylinder:
gi,p,q,k→gi,j,k, where jrepresents the radial distance to the
center of a masked slice.
j
. The domain of pixels in the
maps belonging to the sample was referred to as Ωkwith (i,j)
Figure 2. Illustration of the three data reduction procedures on the
3D segmented volume of gel M4 at the beginning of drying. (a)
Azimuthal integration (GHR maps). (b) Azimuthal + vertical
integration (GR map). (c) Slice integration (GH map). The black
arrows indicate the direction of integration, and the red contours
depict an integrated volume element. The integration step was equal
to the voxel size: 11 μm.
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∈ Ωk. This representation was referred to as the maps of the
gray values along the height and radius (GHR maps).
b) Azimuthal and vertical integration (2D, Figure 2b): the GHR
maps were further integrated over the vertical axis of the
cylinder resulting in a single gray value map along the radius of
the gel and the scan number: gi,j,k→gj,k.j∈ Ωk
rwhere Ωk
ris
the domain of pixels in the map belonging to the sample. This
representation was referred to as the radial gray value map (GR
map).
c) Slice integration (2D, Figure 2c): the masked slices were
integrated over pand q, resulting in a single gray value map
along the height of the gel and the scan number: gi,p,q,k→gi,k.
i∈ Ωk
hwhere Ωk
his the domain of pixels in the map belonging
to the sample. This representation was referred to as the
vertical gray value map (GH map).
A complete description of the reduction procedures can be found
in the SI1. The conversion between a gray value and the
corresponding reconstructed attenuation coefficient (RAC) was
given by
= +
gb a a
255 ( )
(2)
where μis the RAC, and aand bare the custom contrast parameters
used for reconstructing the μCT projections. After data reduction,
RAC maps were generated using eq 2 and referred to as MHR, MR,
and MH maps. These data were then used to model the phase
composition of the drying silica gels.
μCT Drying Model. The spatial and temporal phase composition
of the drying gels was computed by applying a drying model to the
reduced data adapted from our previous study.
10
The construction of
the drying model consists in (1) derive the main equations, (2) make
assumptions to express additional equations, and (3) solve the system
of equations using bilinear interpolation and correction factors. The
model presented in this section applies to the MHR maps resulting
from the azimuthal integration, but the notation can easily be
extended to the two other data reduction procedures. At any time
during drying, the silica gels were composed of three phases: solid
silica skeleton, liquid n-hexane, and vapor/air. In the MHR maps, the
RAC can thus be written as
=
+ +V V V
V
i j k
i j k i j k i j k i j k i j k i j k
, ,
hex, , , hex, , , skel, , , skel, , , air, , , air, , ,
voxel
(3)
where μφ,i,j,kand Vφ,i,j,kare the RAC and volume of a phase φ,
respectively, at a vertical and radial coordinate (i,j) and scan kfor the
hexane, skeleton, and vapor/air phases, and Vvoxel is the volume of a
voxel. The attenuation of the vapor/air phase was set to zero; thus:
μair, i,j,k= 0. The RAC of the silica skeleton and liquid n-hexane
phases was assumed constant and homogeneous throughout drying:
μhex, i,j,k=μhex and μskel, i,j,k=μskel.μhex was calculated from the
reference n-hexane measurements as described in ref,
10
μhex = 0.155.
Eq 3 can be rewritten as a function of the volume fraction of each
phase at a given voxel rather than the total volume, leading to the first
equation of the drying model:
= +f f
i j k i j k i j k
, , hex hex, , , skel skel, , ,
(4)
where fφ,i,j,k=Vφ,i,j,k/Vvoxel for a phase φ. Additionally, the volume
conservation implies
+ + =f f f 1
i j k i j k i j khex, , , skel, , , air, , ,
(5)
The volume fraction maps were referred to as the HEXHR,
SKELHR, and AIRHR maps for fhex, i,j,k,fskel, i,j,k, and fair, i,j,k,
respectively. Solving the system of equations required making
assumptions about the composition of the gels throughout drying.
First, the HEXHR maps were computed by assuming that the hexane
content was zero at the end of drying. To do so, a RAC map
representative of the dry gel was generated:
i
k
j
j
j
j
j
j
j
y
{
z
z
z
z
z
z
z
= + ·
<
N
1
i j k i j k
k k k
i j k k k k, ,
dry
d
, , , ,
F
f f
d f
f f
(6)
where μi,j,kd
f
dry is an artificial MHR map defined over the domain Ωkd
f,kf
= 140 is the final scan, kdis a threshold scan number from which the
hexane content is assumed to be zero, Ndis the number of scans in kd
≤k≤kf,μi,j,k→kd
f
Fis a map generated by bilinear interpolation of a
map from scan k≥kdonto the domain of the map at scan k=kf, and
γk→kd
fis a scaling factor. The scaling factor was defined as the volume
ratio between a source scan k1and a target scan k2as γkd
1→kd
2=Vkd
1/Vkd
2.
Bilinear interpolation was required so that a map defined over a
domain Ω1could match the domain Ω2of another map because the
change in the sample volume throughout drying implied that the
domain Ωkwas different for all scans. The map μi,j,kd
f
dry was then
interpolated from source scan kfonto target scan 0 ≤k< 140,
resulting in the maps μi,j,kd
f→k
dry,F defined over Ωkat any scan. The
HEXHR maps of any scan could be computed by replacing the
product μskel fskel, i,j,kin eq 4 by μi,j,kd
f→k
dry, F with a scaling factor:
=
·
fi j k
i j k i j k k k k
hex, , ,
, , , ,
dry,F
hex
f f
(7)
The SKELHR maps were computed by assuming a zero vapor/air
content at the beginning of drying from scan number k1to k2. The
maps at k1<k≤k2were rescaled onto the domain of target scan k1
and were then averaged over k1≤k≤k2to compute an artificial map
representative of the alcogel μi,j,kd
1
alco :
Ä
Ç
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
Å
É
Ö
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
Ñ
= + +
<
N
1(1 )
i j k i j k
k k k
i j k k k k k k, ,
alco
a, , , ,
F
hex
1 1
1 2
1 1 1
(8)
where μi,j,k→kd
1
Fis a map interpolated from scan k1<k≤k2to scan k
=k1and Nais the number of scans in k1≤k≤k2.kd,k1, and k2were
evaluated using a global quantitative imaging approach and were
reported in ref 10. The rightmost product in eq 8 was an additional
scaling factor. An artificial hexane map representative of the alcogel at
scan k=k1was computed as
=
·
fi j k
i j k i j k k k k
hex, , ,
alco , ,
alco
, ,
dry,F
hex
1
1 f 1 f 1
(9)
The skeleton map at k=k1was then calculated by substituting
fhex, i,j,kby fhex, i,j,kd
1
alco in eq 5, resulting in fskel, i,j,kd
1. The SKELHR maps
for the other scans were computed by interpolating and rescaling the
SKELHR map from scan k1to scan k≠k1:
= ·f f
i j k i j k k k k
skel, , , skel, , ,
F
11
(10)
where
fi j k kskel, , ,
F
1
is the skeleton map interpolated from source scan k
=k1to target scan k≠k1. The AIRHR maps were finally calculated
with eq 5. The complete derivation of the drying model is reported in
SI2 and the bilinear interpolation algorithm is reported in SI3. Sample
M3 had a large meniscus at its bottom (Figure S5), leading to
complications in the data reduction procedure, and was thus
discarded. Similar equations were derived for the MR and MH
maps (not shown), and the resulting volume fraction maps were
referred to as the HEXR, SKELR, and AIRR and HEXH, SKELH and
AIRH maps, respectively. The GR and GH maps and the
corresponding volume fraction maps were interpolated from scan
number to time to create profiles monotonically increasing with time.
Finally, all maps were saved as 2D float arrays, which were then
converted to images to create the figures. The data processing,
modeling and the creation of the figures were carried out in Python
with the DipLib, Matplotlib, NumPy, Pillow, and Scipy libraries.
48−52
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In Operando X-ray Scattering. X-ray scattering experiments
were performed to monitor the structure of the gels during drying at a
wide angle, which corresponds to the molecular structure of the silica
skeleton and n-hexane. The general idea was to correlate the evolution
of scattering intensity in the wide-angle region with the change in the
specimen composition within the volume probed by the beam. X-ray
scattering measurements were performed at the BESSY II synchrotron
of the Helmholtz-Zentrum fur Materialien and Energie (Germany,
Berlin) at the μSpot beamline.
53
One silylated silica gel (labeled M6)
was dried at ambient pressure in a tailored measurement cell adapted
from ref 35. The cell was constructed from anodized aluminum with a
silicon wafer and a silicon nitride window (NORCADA low-stress
SiNx membrane, 10 mm length/width, 1000 nm thickness) placed in
the direction of the X-ray beam. The top of the cell was sealed with a
valve (1/8 in., PN63/1.4408, shortened with an adapter to ca. 26
mm), and a museum glass was placed on the side to allow the
collection of digital pictures of the sample with a digital microscope
camera (TOOLKRAFT USB microscope, 5 MP). The cell was
mounted on a rotary stage that could host up to five cells. At the time
of measurement, sample M6 was transferred from its n-hexane storage
solution to the measurement cell, whereas an empty cell was used for
background correction. The valve was opened fully before the
measurement.
Experiments were performed using a monochromatic X-ray beam
at 18 keV and a B4C/Mo Multilayer (2 nm period) monochromator.
A spot size of 30 ×30 μm2was adjusted by a series of pinholes. The
cell position was set so that the beam hits the sample at a fixed
location, 4 mm from the bottom of the sample. The scattering data
were collected on an Eiger 9 M detector with a 75 ×75 μm2pixel size.
A quartz reference was fixed at the same distance from the beam
source as the sample and was used to determine the sample−detector
distance, beam center, tilt, and rotation. A glassy carbon Standard
Reference Material 3600 (SRM 3600) of the National Institute of
Standards and Technology (NIST) was measured for absolute
intensity calibration.
54
The transmission through the sample was
calculated from the X-ray fluorescence signal collected from a lead
beamstop by using a RAYSPEC Sirius SD-E65133-BE-INC detector
equipped with an 8 μm beryllium window, while the primary beam
intensity was monitored and normalized by using an ion chamber.
Each data frame was collected by exposing the sample to radiation for
1 s every 27 s, with the rotary stage alternating between an empty cell
and the sample cell. The resulting data were preprocessed/previewed
using the DPDAK software package
55
and a custom Python script
utilizing the pyFAI library.
56
The preprocessing steps involved
integration to 1D scattering curves and subtraction of an instrumental
background (i.e., the empty cell). The scattering data were corrected
for transmission and primary beam intensity and corrected for a
“container background”. To normalize the data with the sample
thickness, the diameter of the gel was determined from the optical
images collected during drying (see SI4). As a final step, data were
scaled to absolute units (i.e., cm−1) by sample thickness normalization
and by the scaling factor of the glassy carbon.
The azimuthal integration of X-ray scattering measurements
provided scattered intensity I(q) as a function of the momentum
transfer q= 4πsin (θ/2)/λ, using the wavelength of the synchrotron
beam λand the scattering angle θ, resulting in an accessible qrange of
ca. 0.07 to 40 nm−1. Collected data were analyzed in a wide-angle
diffraction region (3−30 nm−1) to obtain time-dependent volume
fraction profiles. To this end, diffraction data of liquid hexane in a
borosilicate glass capillary were collected. The latter was reduced
following the same procedure described above (i.e., monitor and
transmission normalization, empty capillary subtraction, radial
integration) but not scaled by sample thickness or corrected by
glassy carbon scaling factor.
To compute the hexane, skeleton and vapor/air volume fraction
profiles of the gel at each data frame, the diffraction 1D profiles of the
hexane reference: Ihex(q,t) and of the dry aerogel (i.e., last collected
data frame): Idry(q,t) were modeled using a baseline function and a
function sum of three pseudovoigts:
= +
=
I q t I q a t I q A q r( , ) ( , , ) ( , , , , )
j
jjj j
(base)
1
3
(PV)
0,
(11)
where Iφ(q,t) stands for Ihex(q,t) or Idry(q,t) and the parameters Aj,
q0, j,Γj,rjof the pseudovoigt represent the area, the center, the full
width at half-maximum and the Gaussian−Lorentzian ratio,
respectively. Similarly to the μCT drying model, the hexane content
in the gel at the end of drying was assumed to be zero, so that the
scattered intensity of the dry gel arose only from the skeleton and
vapor/air. The baseline function for the hexane reference: Ihex
(base)(q,a,
t) was set as a linear polynomial with slope a:
= ·I q a t a t q( , , ) ( )
hex
(base)
(12)
and the baseline function for the dry aerogel was set as a power law
decay with a constant arepresenting the law’s exponent:
= =I q a t t q( , , ) a t
dry
(base)
f
( )
f
(13)
where t=tfis the time of the last data frame. During the modeling of
the hexane and dry aerogel diffraction profiles, an additional
parameter representing the data background was used. To obtain
volume fraction profiles of each phase, two more steps were necessary.
At first, the volume fraction of the skeleton in the dry aerogel was
calculated by assuming a composition: Si23O40C9H28 and a skeletal
density of approximately 1.9 g cm−3. That composition was estimated
by comparing the weight of fully dried silylated gels with fully dried
unmodified gels
10
and by assuming that the weight difference was
only due to the silyl groups in the modified gels: Si(CH3)3. Moreover,
the unmodified gels were left in a desiccator for 24 h before being
weighed to complete drying.
Subsequently, by means of the Python library xraylib
57,58
and the
Beer−Lambert equation, it was possible to compare the experimental
transmission of the dry aerogel Tskel and the volume fraction-
dependent theoretical transmission:
=T f dexp( )
dry skel
dry
skel dry
(14)
with
fskel
dry
the skeleton volume fraction of the dry gel, μskel its
attenuation coefficient for an 18 keV X-ray beam, and ddry the sample
diameter obtained with the optical microscope at the corresponding
time. The model functions Ihex(q), Idry(q) and
fskel
dry
were combined
and used for fitting the time-dependent scattering profiles I(q,t) of
the drying gel as follows:
= + +I q t s f t I q f t
fI q b t( , ) ( ) ( ) ( ) ( ) ( )
hex hex hex skel
skel
dry dry
(15)
with fhex(t) and fskel(t) the hexane and skeleton time dependent
volume fractions, b(t) a background independent of q, and shex a
constant factor to scale the hexane data to absolute units (i.e., cm−1).
shex was calculated by assuming that at t= 0, the vapor/air content in
the gel was zero, leading to
= = + =s f t f t1 ( 0) ( 0)
hex hex skel
(16)
At t> 0, the time dependent vapor/air volume fraction fair(t) was
calculated by modifying eq 16:
= +f t s f t f t( ) 1 ( ( ) ( ))
air hex hex hex
(17)
which implies that fair(t= 0) = 0. All data modeling was performed
using the Scipy optimize library provided by Python.
52
In order to
improve performance and result reliability, fits were performed by
providing analytical functions and jacobians. Best-fit parameter
uncertainties were therefore calculated by evaluating the Jacobian at
the minimum of the penalty function distribution.
Figures were generated using the scientific color maps batlow,
lapaz, and oslo
59
to prevent visual distortion of the data and exclusion
of readers with color-vision deficiencies.
60
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■RESULTS AND DISCUSSION
Hexane and Vapor/Air Spatial and Temporal Dis-
tributions. This section reports and discusses the spatial
distribution and evolution of the gel phase composition
generated by μCT quantitative imaging. The azimuthal
integration of the masked slices produced well-defined GHR
maps shown in Figure 3a for sample M4 at different drying
stages (see Video S1 for an animation of the 141 frames). The
drying shrinkage can be seen with the decrease in height and
maximum radius of the gel up to the maximum shrinkage at 7.6
h, together with an increase of the average gray values. The
maximum shrinkage was followed by the re-expansion of the
gel (spring-back effect) and by a decrease in the gray values.
Silica gels dried in the fume hood under similar conditions
remained transparent until the maximum shrinkage and only
turned opaque upon re-expansion. The hexane, skeleton, and
vapor/air volume fractions along the height and radius of
sample M4 are shown in Figure 3b−d and in Video S1. At the
start of drying, the hexane and skeleton phases were uniformly
distributed in the gel at an average volume fraction of 94 and 6
vol %, respectively, suggesting a homogeneous gelation
process. The fraction of vapor/air was 0 vol % as it was set
in the drying model. Up to the maximum shrinkage, the hexane
content decreased while the skeleton and vapor/air contents
increased, each phase being still relatively homogeneously
distributed across the entire gel’s volume. The emergence of
the vapor/air phase in the gel was not clearly depicted in the
AIRHR maps due to noise in the data, but Figure 3d shows a
non-negligible amount of vapor/air before the maximum
shrinkage, as already reported in ref 10. Near the end of drying,
Figure 3. GHR and volume fraction maps of sample M4 at 10 selected drying stages on top of a cyan background. (a) GHR maps with the
corresponding gray value scale on the right. The brightness and contrast in the images of the GHR maps are adjusted to improve visualization. (b)
HEXHR maps. (c) SKELHR maps. (d) AIRHR maps. The color scale of the volume fraction maps is shown at the bottom right of the figure. The
volume fraction maps are normalized between 0 and 100%. The images of the volume fraction maps are encoded with a gamma value of 0.5 to
improve the visualization. The time scale is illustrated with an arrow on top of the figure, and the time gap between the maps in a given panel is 1.56
±0.05 h. The length scale of all maps is indicated in the first map of panel (d). The maps corresponding to the maximum shrinkage are outlined in
red. Each map consists of 410 ×1455 noninterpolated data points.
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the skeleton and vapor/air spatial distributions were
homogeneous, and after 14 h of drying, the gel was composed
of about 22 and 78 vol % of skeleton and vapor/air,
respectively. Heterogeneities in the repartition of hexane and
vapor/air appeared at the maximum shrinkage where the
hexane volume fraction abruptly dropped to ca. 0 vol % at the
top of the gel (Figure 3b, Video S1). However, the fact that
each map corresponds to a single drying stage and the presence
of noise in the maps limited the analysis. The spatial and
temporal distribution of the hexane and vapor/air phases was
Figure 4. Radial and vertical maps of the gray values and volume fraction of sample M4 on top of a cyan background. The 3D image on the left of
the figure depicts the segmented volume of M4 at the beginning of drying, and the dashed lines illustrate the radial and vertical axes of the cylinder
against which the radial and vertical maps are shown. (a) GR and GH maps. The gray value scale is shown at the bottom of panel (a). The
brightness and contrast in the images of the GR and GH maps are adjusted to improve visualization. (b) HEXR and HEXH maps. (c) SKELR and
SKELH maps. (d) AIRR and AIRH maps. The images of the volume fraction maps are encoded with a gamma value of 0.5 to improve visualization.
The time axis is shown in each vertical map, and the length scale is shown in the radial and vertical maps of panel (a). The radial maps consist of
1460 ×410 data points and the vertical maps consist of 1460 ×1455 points. In all maps, the horizontal time resolution is interpolated from 141
time stamps onto 1460 points.
Figure 5. Volume fraction profiles of hexane and vapor/air along the height of sample M4 at selected time stamps. Vapor/air profiles were between
6.9 and 7.8 h (a) and between 7.6 and 8.5 h (b). Hexane profiles between 6.9 and 7.8 h (c) and between 7.6 and 8.5 h (d). The profiles in panels
(a) and (c) correspond to the μCT scans before the maximum shrinkage and shortly after, while the profiles in panels (b) and (d) correspond to
the μCT scans at the maximum shrinkage and after. The dashed lines correspond to the bottom (h= 0 mm) and the top of the sample. The time of
maximum shrinkage is highlighted in red in the legends. The spacing between the profiles in each panel corresponds to a single μCT scan. The
black arrows depict the drying time. The profiles were extracted from the AIRH and HEXH maps by excluding the values affected by the artifacts at
the edges for better visualization.
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thus analyzed based on the vertical and radial volume fraction
maps derived by modeling the GH and GR maps, which had
the advantage of depicting the gel state at all drying stages
along the two main axes of the cylindrical samples. The results
are shown in Figure 4.
The noise in the gray values of the vertical and radial maps
in Figure 4a was significantly lower compared to the
corresponding GHR maps, resulting in smoother volume
fraction maps, which permitted a more accurate evaluation.
Slight variations in the vertical distribution of hexane were
observed before the maximum shrinkage, with a higher
concentration of hexane at the bottom of the gel than at the
top (Figure 4b). The top of the gel was composed of relatively
more vapor/air than at the bottom before the maximum
shrinkage (Figure 4d), whereas the skeleton spatial distribution
remained static throughout the entire drying process, as set in
the drying model (Figure 4c). Upon spring-back, significant
variations in the repartition of hexane and vapor/air appeared
along the height of the gel. The top region of the gel got
depleted in hexane first, leading to an increase in the vapor/air
volume fraction in the same region (Figure 4b,d). Hexane and
vapor/air vertical profiles were extracted from the HEXH and
AIRH maps to quantify the variations before and after
maximum shrinkage (Figure 5). At the maximum shrinkage
(7.6 h), the vapor/air content abruptly increased from 30 to 52
vol % near the top of the gel within 6 min (Figure 5a) and was
followed by a wave-like drying front traveling downward in the
sample as the gel started re-expanding (Figure 5b). At 7.9 h,
the vapor/air content rose at the bottom of the gel which
corresponded to another drying front traveling upward. At 8.5
h, those heterogeneities along the gel height stabilized,
although a gradient of vapor/air was still present from the
top to the bottom of the gel (Figure 5b). Similar but opposite
features were noted in the hexane vertical profiles (Figure
5c,d). It was worth noting that at 8.5 h, there was still a
significant fraction of hexane remaining with up to 29 vol % in
the bottom regions of the gel (Figure 5c,d). At ca. 10 h of
drying, the re-expansion of the gel slowed down and the
variations in the hexane and vapor/air content along the gel
height dissipated with an average hexane volume fraction close
to zero and an average vapor/air volume fraction at 73 vol %.
Between 10 and 14 h, the volume fraction of vapor/air slightly
increased as the gel re-expanded to reach a final value of 78 vol
% (Figure 4d).
The emergence of the vapor/air phase in the gels was
referred to as the “cavitation onset” and was evaluated from the
AIRH maps by extracting vertical volume fraction profiles near
the start of drying (Figure 6). The vertical profiles indicated a
nonzero amount of vapor/air from 2.0 h of drying in sample
M4, with a higher concentration in the middle of the gel.
Similar conclusions were drawn from the analysis of samples
M1, M2, and M5. The only notable difference was a shift in the
timing and duration of specific events such as the spring-back
effect and the emergence of the vapor/air phase (especially in
sample M1) due to slightly different starting volumes of the
gels and possibly drying conditions (Figures S9−S18). As an
attempt to quantify the cavitation onset, the AIRH maps
(Figures 4d and S12d−S14d) were integrated over the height
of the gel, giving an average vapor/air volume fraction. The
cavitation onset was then defined as the drying time at which
the average vapor/air volume fraction rose above an arbitrary
volume fraction of 1 vol %. The time of cavitation onset for
samples M1, M2, M4, and M5 was 2.93, 3.50, 3.64, and 3.43 h,
respectively. Additionally, the pore volume shrinkage was
calculated at the cavitation onset. It was defined as νp,kd
cav =
Vp,kd
cav/Vp,0 where Vp,kd
cav, and Vp,0 are the pore volume at the
cavitation onset and the start of drying, respectively. The pore
volume shrinkage was 64.1, 55.3, 52.7, and 51.8 vol % for
samples M1, M2, M4, and M5, respectively. The time and pore
volume shrinkage of sample M1 seemed to deviate compared
to those of the three other samples. The global quantitative
imaging analysis on that sample also showed deviations
compared to the other samples.
10
This sample put apart, the
proximity of the results between the three other samples
suggested that the cavitation onset was related to a particular
state of the gel. Figures 5a and 6showed a steady growth of the
fraction of the vapor/air phase in the gel from the cavitation
onset until the maximum shrinkage throughout drying, which
was also observed in the other samples. Table 1 summarizes
the properties of the gels upon the cavitation onset.
To corroborate the observations made from the μCT
measurements, the phase composition derived by modeling of
the WAXS data was compared with the results from μCT
modeling at a representative location in the gel. Although
WAXS did not allow spatial resolution of the vapor-air phase, it
could detect the emergence of a vapor-air phase prior to the
maximum shrinkage. Figure 7 shows the hexane, skeleton, and
vapor/air volume fraction profiles computed from both
methods against a normalized time scale. The evolution of
the scattering profiles in the 3−30 nm−1region and examples
of the data fit can be found in Figures S19 and S20,
Figure 6. Vertical vapor/air profiles extracted from the AIRH maps of
sample M4 between 2.0 and 2.9 h. The dashed lines correspond to the
bottom (h= 0 mm) and the top of the sample. The spacing between
each profile corresponds to a single μCT scan. The black arrow
depicts the drying time.
Table 1. Properties of All Samples at the Cavitation Onset
a
sample tcav
(h) tcav/tMS
pore volume shrinkage at
tcav (vol %) volume shrinkage at
tcav (vol %)
M1 2.93 0.360 64.1 66.5
M2 3.50 0.436 55.3 58.8
M4 3.64 0.471 52.7 55.8
M5 3.43 0.482 51.8 54.9
a
tcav and tMS stand for the time of cavitation onset and the time of
maximum shrinkage, respectively. The volume shrinkage is the ratio of
the gel volume to the gel initial volume at the start of drying.
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respectively. The volume fraction profiles corresponded to the
composition of the gels 4 mm above their bottom (where the
X-ray beam probed the sample during the in operando X-ray
scattering experiment). The profiles generated by these two
methods were relatively consistent. The WAXS modeling
results suggested a content of 14 vol % of vapor/air in the gel
before the maximum shrinkage, supporting the μCT results.
The initial and final compositions of the gels were similar
between both methods with variations of ca. 2 vol %. In the
probed location, the fraction of vapor/air raised above 1 vol %
at t/tMS ≈0.3 from the μCT measurement, and at t/tMS ≈0.63
in the WAXS measurements (Figure 7). This cavitation onset
in the gels dried in the μCT setup was lower than the values
reported in Table 1 because it only accounted for the gas
volume detected at a specific height in the gels. The differences
observed between μCT and WAXS results during drying could
be due to the different environments in the two experiments.
Sample M6 dried faster in the scattering setup and reached the
maximum shrinkage after ca. 4.3 h of drying versus ca. 7.5 h for
the samples dried in the μCT setup. This could suggest that
the gel composition and the cavitation onset have a nonlinear
dependency on the drying rate. Additionally, the lower
skeleton volume fraction from WAXS modeling compared to
μCT modeling could be related to the assumptions on the
chemical composition and density of the silica skeleton set in
the WAXS drying model.
Evaporation Mechanisms. We interpreted the spatial and
temporal phase composition of the gels in terms of evaporation
mechanisms, with an emphasis on cavitation. Based on the
distribution and evolution of the hexane and vapor/air phases,
we propose that the evaporative drying of the silica gels
prepared in this study proceeded in three distinct stages:
(1) Evaporation by drying shrinkage from 0 to ca. 3.5 h.
(2) Evaporation by a combination of cavitation and drying
shrinkage from ca. 3.5 to 7.5 h (maximum shrinkage).
(3) Evaporation by meniscus recession from ca. 7.5 h until
complete evaporation of the remaining hexane.
During stage (1), the hexane content decreased homoge-
neously without any vapor/air inclusions (Figure 4d) and the
volume of hexane evaporated matched the volume shrinkage of
the gel, which was consistent with the drying shrinkage
model.
15,20
During stage (2), the drying shrinkage proceeded
until the maximum shrinkage was reached in parallel with the
growth of a vapor/air phase in the gels. Both the μCT and
WAXS results showed that vapor/air appeared ahead of the
maximum shrinkage and increased steadily, and μCT volume
fraction maps revealed a relatively homogeneous distribution
of vapor/air across the whole gel’s volume (Figures 3d, 5a, and
6), which suggested it was created by cavitation of hexane. In
stage (3), the gel sprung back and a drying front was observed
growing vertically through the sample (mostly downward).
This stage was associated with the recession of the hexane-
vapor interface into the pores and corresponded exactly to the
re-expansion of the gels. Gels dried in the fume hood turned
opaque upon re-expansion and gradually shifted toward a
bluish color under a dark background, the latter being caused
by Rayleigh scattering.
61
The time resolution of the μCT
experiments did not allow us to distinguish potential adiabatic
burst events. Presumably, no additional cavities were created
ahead of the drying front in stage (3) as the vapor/air volume
fraction did not increase at those locations (Figure 5a). The
gradient in the hexane and vapor/air volume fraction along the
height of the gel during the re-expansion confirmed the
heterogeneous nature of the spring-back effect which was
attributed to the design of the drying chamber used in this
study and in ref 10.
The hypothesis of cavitation was indirectly supported by two
experimental observations. First, the vapor-air phase grew
uniformly without any visible drying front. This was
inconsistent with an evaporation mechanism by recession of
the meniscus, which would proceed by the ingress of the
liquid−vapor interface into the pores, forming a drying front. It
also seemed unfeasible that evaporation by drying shrinkage
and meniscus recession occurred simultaneously in the gels,
given that the recession of the meniscus would result in local
relaxation of the capillary stress, whereas the drying shrinkage
kept proceeding steadily throughout stage (2). The absence of
a drying front supports evaporation by cavitation although it
must be noted that this feature might also be associated with
other mechanisms, such as fractal-like penetration of the vapor
phase.
62,63
Second, the gels turned opaque only upon re-
expansion and not at the cavitation onset. The change of
transparency during stage (3) arguably corresponded to the
apparition of pores filled with vapor/air that showed a
characteristic size large enough to scatter visible light (400−
700 nm). As such a change was not observed during stage (2),
it suggested the absence of a continuous medium of vapor/air
in the gels larger than 400 nm. This observation would be
consistent with the growth of vapor/air within the gels by
cavitation with cavities smaller than the wavelength of visible
light. Nevertheless, it may be possible for gels to turn opaque
upon cavitation.
40−42
Those elements suggested that the
vapor/air phase in the gel prior to the maximum shrinkage
was not caused by meniscus recession but by cavitation of the
solvent. Additionally, the cavitation onset coincided with a
pore volume shrinkage of 51−64 vol % (Table 1). Because
cavitation in porous materials depends on the porous media
stiffness,
23,25
which is closely related to the pore volume, the
pore volume shrinkage is a critical parameter for the emergence
of cavitation in the gels. As vapor/air bubbles appeared at the
start of stage (2), they grew steadily and uniformly as shown by
the vapor/air vertical distribution (Figures 4d, 5a, and 6). This
growth was consistent with the phenomenology of evaporation
by cavitation and drying shrinkage. The first cavities were
Figure 7. Average phase composition of gel M6 computed by WAXS
modeling (full lines) along with the phase composition computed by
μCT modeling, averaged over gels M1, M2, M4 and M5 (dashed
lines) at the same location in the gel. The time scale is normalized by
the time of maximum shrinkage tMS, which was 4.3 h for the WAXS
modeling and ca. 7.5 h for the μCT modeling.
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presumably created in large pores ahead of the outer surface of
the gel, where the liquid confinement by the solid matrix was
the smallest.
23
At this stage, there was no deceleration of the
volume shrinkage upon the emergence of vapor/air in the
gel.
10
As the drying shrinkage proceeded, the tension in the
liquid rose, which enabled the nucleation of additional cavities
in smaller pores and possibly the growth of already-created
bubbles. Shortly before maximum shrinkage, the volume
shrinkage decreased,
10
which could indicate a stress release
by cavitation.
To assess whether cavitation can actually occur in silica gels
filled with hexane, the pressure and required pore size were
estimated following the arguments of ref 64. The energy
change associated with the creation of a spherical cavity of
radius Ris 4πR3pl/3 + 4πR2γ, with pl< 0 the pressure in the
liquid.
14,65
A bubble with a critical radius larger than R*=
−2γ/plwill spontaneously grow, and the corresponding energy
barrier is ΔE= 16πγ3/3pl
2.
65
Cavitation can occur only in pores
larger than R*. According to the classical nucleation theory, the
nucleation rate is
64
=J J e E k T
0
/B
(18)
where J0is a prefactor in cm−3s−1,kBis the Boltzmann
constant, and Tis the temperature. Silica gels display a fractal
structure
66
with a wide size distribution of pores (1−100 nm).
3
In silica gels, cavitation will occur in sufficiently large pores if
the pressure generated at the meniscus (in smaller pores within
the network) is large enough to generate a reasonable
nucleation rate, as shown in eq 18. At the meniscus, the
pressure in the liquid is given by eq 1:pl=p0−2γ/rassuming
that hexane fully wets the gel (θ= 0), where ris the smallest
pore radius blocking the recession of the meniscus in the gel.
In nanometric pores, |pl| ≫ |p0|and the vapor pressure can be
neglected.
39
The energy barrier for the nucleation of a bubble
can be rewritten as a function of the smallest pore radius: ΔE=
4πγr2/3. Ref 64 provides an estimate for the prefactor J0for
cavitation conditions:
=J N M2 /
0
2
A
3
w
3
, with ρbeing the
liquid density, NAbeing Avogadro’s constant, and Mwbeing
the molar mass of the liquid.
The nucleation rate of hexane in silica gels was then
numerically estimated as a function of the smallest pore radius
in the gel using eq 18. The dependence of n-hexane surface
tension on the temperature was taken into account using a
modified van der Waals equation reported in ref 67. The
results are shown in Figure 8. The nucleation rate showed an
extreme dependence on the radius of the smallest pores. The
liquid temperature also has a strong influence on the
nucleation rate, promoting cavitation at higher temperatures
as already reported elsewhere.
26,27,29
To evaluate what would be a reasonable nucleation rate for
cavitation in silica gels, the numerical estimates of Jwere
compared with the experimental growth rate of vapor/air from
the μCT analysis. The growth rate per unit volume and per
unit time was calculated as Jexp(r) = Δfair/(Δt·4πr3/3), where
Δfair is the difference of the average volume fraction of vapor/
air in gel M4 taken between tcav and tMS, with tcav the time of
cavitation onset and tMS the time of maximum shrinkage, and
Δt=tcav−tMS = 4.1 h. Δfair/Δtwas a good approximation of the
vapor/air growth rate (Figure S21). The profile Jexp(r)
corresponds to the creation of spherical vapor bubbles of
radius rthat neither grow nor collapse during the time lapse
Δt. The intersection between the profiles Jfrom the CNT
estimations and Jexp in Figure 8 gives an approximation of the
nucleation rate that would correspond to the growth rate of the
vapor/air phase in gel M4. At 25 °C (average temperature in
the μCT instrument),
10
a nucleation rate of about 2 ·1015
cm−3s−1was found for a pore radius blocking the meniscus
recession of r= 1.47 nm. Under these conditions, the liquid
pressure would be pl≈−24 MPa, which was in good agreement
with the values found by optical measurements of hexane
desorption in alumina membranes.
22
These results supported
that hexane cavitation can occur in silica gels, which have a
small enough pore size for the liquid to reach the negative
pressure required to form hexane bubbles at a reasonable rate.
However, those results did not consider the dynamic of hexane
bubbles over time. The spontaneous growth of already formed
bubbles could notably produce the vapor/air volume in sample
M4 at a much lower nucleation rate; thus, cavitation might also
occur at larger pore radii. On the other hand, the hexane
temperature during drying was likely lower than 25 °C due to
the latent heat of vaporization, and this would reduce the
nucleation rate at a fixed pore radius.
The timing and extent of cavitation could largely differ in
gels resulting from different synthesis routes, depending on the
porous network, stiffness of the solid matrix, and the nature of
the solvent. Notably, in certain systems, cavitation may be
absent. Cavitation in the silica gels synthesized in this study
could have been facilitated by the presence of a thicker layer of
silica on the outer surface of the gels, which may have formed
during gelation in the molds. Such a shell with a smaller pore
size would have a pore-blocking effect,
28
generating a stronger
capillary pressure during drying that enables nucleation of
vapor/air bubbles ahead of the gel surface. Additionally,
cavitation of solvents with a low surface tension (such as
hexane) is facilitated since the free energy of cavitation bubbles
is proportional to γ3at a given liquid pressure.
14,64
As shown
by the CNT estimations, the temperature at which hexane
evaporation occurs has a significant influence on the cavitation
rate. Performing ambient-pressure drying at higher temper-
atures may therefore reduce the drying stress and possibly
allow the production of larger monolithic aerogels with higher
spring-back efficiencies, which may be investigated in the
future. To our knowledge, the effect of the drying temperature
on the spring-back efficiency and size of monolithic aerogels
Figure 8. Calculated rate (full lines) of the formation of hexane
bubbles Jas a function of the smallest pore radius rblocking the
recession of the meniscus at different liquid temperatures. Vapor/air
growth rate (dashed line) was calculated from the μCT experiments
on sample M4, assuming the creation of spherical bubbles of radius r.
The black arrow depicts the temperature increase.
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has not been investigated, though sometimes temperatures of
50 °C and more have been used. The surface chemistry of the
silica network may also play a role: defects on the silica surface
(e.g., remaining silanol groups due to incomplete silylation)
can enable heterogeneous nucleation of vapor/air bubbles
which require less energy than homogeneous nucleation,
31,68
meaning that cavitation can occur at a reduced liquid pressure.
However, recent investigations in porous silicon suggest that
desorption exclusively takes place through homogeneous
cavitation in that system.
22,26
This is attributed to the complete
wetting of silicon and most surfaces by hexane, liquid nitrogen,
and liquid helium, owing to their exceptionally low surface
tension. Cavitation in silica gels could also be detected by
complementary measurements and analyses, such as acoustic
measurements
69−71
and Small-Angle X-ray Scattering
(SAXS),
72
which were not conducted in this work.
Reliability and Limitations. This section discusses the
reliability of the μCT data reduction procedures and modeling
and lists the artifacts generated experimentally and computa-
tionally. The GR maps of all samples indicated a heteroge-
neous distribution of the gray values along the gel radius at all
drying stages (Figures 4a and S12a−S14a). This feature was
also visible in the GHR maps to a lesser extent (Figures 3a and
S9a−S11a). Those variations were more pronounced near the
maximum shrinkage. A spatial variability analysis revealed an
exponential dependency of the gray values on the gel radius
(SI5). The relative increase of the gray values along the radius
was correlated to the gel diameter and attributed to beam
hardening,
43
although potential heterogeneities in the skeleton
concentration along the gel radius could not be excluded. A
spatial variability analysis on the gray values across the gel’s
height showed no sign of beam hardening. A similar analysis of
the distribution of the gray values along the azimuth of the
cylinders revealed slight variations that were caused by
reconstruction artifacts rather than being a physical feature
of the samples (SI5). Darker lines can be seen on the top and
bottom of the gel and its outer radius in the GHR maps
(Figure 3a). These were caused by background inclusions in
the masked images due to imperfect segmentation. Locally
higher gray values were observed at a height of ca. 0.7 mm
from the bottom of the gels (Figure 3a), which were
reconstruction artifacts due to the proximity of the sample to
the bottom of the PEEK drying chamber and the low number
of CT projections used for reconstruction. The artifacts in the
GHR maps were propagated to the GR and GH maps, where
similar features were observed at the vertical and radial edges
of the samples. Despite these artifacts, the overall kinetics of
the gray values in the GHR, GR, and GH maps were consistent
with the results of the global quantitative imaging analysis
(SI6). This confirmed the reduction procedures could generate
accurate representations of the shape-changing samples. The
histograms of the GHR maps also showed a narrower
distribution of the gray values compared to the histogram of
the masked slices, as a result of the azimuthal integration
(Figure S31). Lastly, the gray values in the raw reconstructed
slices were affected by the anode heel effect,
47
which was
corrected by the method developed in ref 10.
The volume fraction maps were affected by the artifacts in
the gray value maps. The abnormally high/low volume fraction
of a given phase at the edges of the gel (Figures 3b−d and 4b−
d) corresponded to the locations where the segmentation
included the background in the masked slices, which was
interpreted in the drying model as a change in the
composition. The abrupt changes at the top, bottom, and
radial edges of the gels were thus treated as artifacts. The radial
variations of the gray values were also propagated to the
volume fraction maps, which can be seen, for example, in the
increasing vapor/air volume fraction with the gel radius in the
AIRR map of sample M4 before the maximum shrinkage
(Figure 4d). Those variations, being partially attributed to
beam hardening, prevented making reliable interpretations of
the radial distribution of hexane, skeleton, and vapor/air
during drying. The vertical maps were the most reliable
representations to quantify the phase composition of the gels
as they appeared to be free of beam hardening artifacts.
Nonetheless, the HEXHR, SKELHR, and AIRHR maps
depicted qualitative aspects of the composition evolution
during drying despite the artifacts in the radial direction. The
kinetics of the hexane, skeleton, and vapor/air volumes were
consistent with the results from the global quantitative imaging
approach,
10
besides some irregularities near the maximum
shrinkage that were possibly related to the additional
computational steps required in the presented method (SI6).
At the beginning of stage (3), as the drying front emerged
on the top of the gel, an increase in the hexane concentration
was observed in regions ahead of the drying front in all samples
(Figures 5c and S16c−S18c). A similar but opposite feature
was observed in the vapor/air phase with a decrease in the
vapor/air volume fraction at those locations. This phenomen-
on can also be seen in Figure 7, where the hexane and vapor/
air volume fractions were computed at a constant height in the
gel. Most likely this phenomenon was related to an artifact
arising from the assumption of a temporally static skeleton
distribution throughout drying. This assumption seemed valid
for most of the drying process, as the volume shrinkage
appeared to be uniform along the height and radius of the gel
(Figure 3), but may present limitations at the onset of stage
(3) due to the heterogeneous nature of the spring-back effect
in this study. The HEXHR maps were computed based on an
artificial MHR map representative of the dry gel, which was
rescaled toward a target scan. That artificial MHR map was
stretched by bilinear interpolation to fit the target scan domain
and corrected by a scalar scaling factor given by the volume
ratio between the source scan and the target scan. The overall
volume of the gel increased at the spring-back, which decreased
the rescaling factor γkd
f→kused to compute the local hexane
volume fraction (see eq 7), resulting in an effective increase of
the local hexane volume fraction that did not account for the
heterogeneous re-expansion of the gel. This also resulted in an
overall decrease of the vapor/air volume fraction ahead of the
drying front, as it was calculated from eq 5. Nevertheless, it
could not be excluded that part of the observed variations were
a physical feature of the sample. The sudden relaxation of the
liquid tension could notably destabilize the equilibrium of the
vapor/air bubbles close to the drying front and possibly result
in a collapse of the cavities. This would increase again the
tension in the liquid, which may pull more hexane ahead from
the drying front, resulting in an effective increase of the hexane
volume fraction at the corresponding location. Including the
local re-expansion of the gel in the bilinear interpolation
routine would require generating a dynamic mesh depending
on both the shape changes and local gray values, which was out
of the scope of this study.
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■CONCLUSIONS
This study addressed the evaporation mechanisms during the
evaporative drying of silica gels by evaluating the spatial and
temporal phase compositions modeled by μCT quantitative
imaging. A noteworthy discovery was made, as the observed
evolution of vapor/air content in the gels aligned with the
concept of evaporation by cavitation initially theorized by
Scherer and Smith in 1995. The presence of vapor and air in
the gels before the maximum shrinkage was confirmed by
WAXS modeling. The repartition of hexane and vapor/air in
the gels was successfully computed by an in operando μCT
workflow, which also demonstrated the potential of μCT
quantitative imaging to generate local phase composition maps
of the shape and composition of evolving materials. Based on
these results, we proposed that the evaporative drying of
silylated silica gels proceeded in three stages: (1) evaporation
by drying shrinkage; (2) evaporation by drying shrinkage and
by cavitation; (3) evaporation by the recession of the
meniscus, challenging the common drying model associated
with sol−gel processes. By using classical nucleation theory, we
have also derived the nucleation rate and smallest pore radius
required to create the vapor/air volume computed by μCT
quantitative imaging, which supported that cavitation can
occur in silica gels. The emergence of cavitation was correlated
with a pore volume shrinkage of about 50 vol % that was
attributed to the critical point where the silica matrix stiffened
enough to enable the nucleation of cavities. Cavitation started
as early as 3.4 h of drying time, whereas the maximum
shrinkage occurred at 7.6 h of drying.
In general, this discovery highlights cavitation as a new
potential mechanism for evaporation in silica gels and makes a
valuable contribution to understanding drying processes in
porous materials. Tailoring the evaporative drying process by
cavitation could be advantageous in the production of
monolithic aerogels as it would reduce the pressure gradients
in the gels and alleviate the rise of the capillary pressure, which
would in turn reduce the risk of cracks appearing during
drying. This could be done by precise modulation of gel
properties (specifically pore size and matrix stiffness) and
optimization of drying conditions and may represent a
promising route for substantial advancements in the fabrication
of monolithic aerogels through ambient-pressure drying, a
process currently constrained in its application. The study
notably highlighted the potential of carrying out evaporative
drying at temperatures higher than room temperature, which
may promote cavitation. The characterization of cavitation in
the drying process of silica gels could be expanded through the
application of alternative nondestructive methodologies,
including acoustic detection and SAXS.
■ASSOCIATED CONTENT
Data Availability Statement
The raw data, processed data, python scripts and intermediate
results are openly available and are archived on Edmond
73
https://doi.org/10.17617/3.OYI3T.
*
sı Supporting Information
The Supporting Information is available free of charge at
https://pubs.acs.org/doi/10.1021/acs.langmuir.4c00497.
μCT data reduction procedure, derivation of the μCT
drying model, bilinear interpolation algorithm, artifact in
sample M3, determination of the gel diameter during in
operando X-ray scattering measurements, volume
fraction maps and profiles of samples M1, M2, and
M5, time series and fitting of WAXS data, gradient of
vapor/air average volume fraction, spatial variability
analysis of μCT data, comparative analysis of μCT
quantitative imaging approaches, histograms of masked
slices and GHR maps (PDF)
Video of the distribution of the gray values and hexane,
silica skeleton, and vapor/air volume fractions along the
gel’s height and radius for sample M4 (MP4)
■AUTHOR INFORMATION
Corresponding Authors
Julien Gonthier −Department of Biomaterials, Max Planck
Institute of Colloids and Interfaces, 14476 Potsdam,
Germany; orcid.org/0000-0001-5257-4688;
Email: [email protected]
Wolfgang Wagermaier −Department of Biomaterials, Max
Planck Institute of Colloids and Interfaces, 14476 Potsdam,
Germany; Email: [email protected]
Authors
Ernesto Scoppola −Department of Biomaterials, Max Planck
Institute of Colloids and Interfaces, 14476 Potsdam,
Germany; orcid.org/0000-0002-6390-052X
Tilman Rilling −Department of Biomaterials, Max Planck
Institute of Colloids and Interfaces, 14476 Potsdam,
Germany
Aleksander Gurlo −Chair of Advanced Ceramic Materials,
Institute of Materials Science and Technology, Faculty III
Process Sciences, Technische Universität Berlin, 10623 Berlin,
Germany; orcid.org/0000-0001-7047-666X
Peter Fratzl −Department of Biomaterials, Max Planck
Institute of Colloids and Interfaces, 14476 Potsdam,
Germany; orcid.org/0000-0003-4437-7830
Complete contact information is available at:
https://pubs.acs.org/10.1021/acs.langmuir.4c00497
Author Contributions
Conceptualization: J.G., E.S., P.F., and W.W.; methodology:
J.G. and E.S.; software: J.G. and E.S.; formal analysis: J.G., E.S.
and P.F.; investigation and data curation: J.G., E.S., and T.R.,
writing−original draft: J.G., E.S., and W.W.; writing−review
and editing: J.G., E.S., T.R., A.G., P.F., and W.W.; visualization:
J.G.; supervision: P.F. and W.W.; project administration: J.G.,
P.F., and W.W.; funding acquisition: A.G. and W.W.
Funding
This project was funded by the Deutsche Forschungsgemein-
schaft (DFG, German Research Foundation)�No.
454019637. Open access funded by Max Planck Society.
Notes
The authors declare no competing financial interest.
■ACKNOWLEDGMENTS
We thank F. Zemke for the support and valuable discussions,
C. Li for his assistance at the μSpot beamline, M. Bott and T.
Schmidt for designing and manufacturing the PEEK molds and
drying chambers, S. Valton from RX Solutions for her input on
the reconstruction of the μCT scans, D. Werner for his help on
the μCT instrument, and D. Friese for his participation on the
artwork. We acknowledge beamtime at the μSpot beamline
(HZB proposal 222-11609-CR) of Helmholtz-Zentrum Berlin
fur Materialien und Energie, Berlin, Germany. We also thank
Langmuir pubs.acs.org/Langmuir Article
https://doi.org/10.1021/acs.langmuir.4c00497
Langmuir 2024, 40, 12925−12938
12936
the Deutsche Forschungsgemeinschaft (DFG) for supporting
this research.
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