Solvent Cavitation during Ambient Pressure Drying of Silica Aerogels

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.


SI1: Data Reduction Procedure
The in operando µCT measurements generated a 4D dataset as 3D reconstructed volumes over time for each sample.This section describes the three reduction procedures used to reduce the 4D dataset.Note that in the main text, the notation was simplified and the 4D dataset was introduced as  ,,, , describing it as the gray values already corrected for the anode heel effect.In practice, the anode heel effect correction was performed after a preliminary integration.
The 4D gray values were referred to as  � ,,, , with , , ,  ∈ ℕ.The tilde stands for uncorrected, the index  depicts the vertical position in the sample in voxels, the indexes  and  depict the horizontal positions in voxels and the index  stands for the scan number (time).Figure S1 illustrates the three spatial axes and the time axis.Note that the index  was defined pointing downwards following the convention for digital pictures.The 4D dataset consisted of a series of images (the masked slices) indexed by  and .As a reminder, the masked slices are 8-bit tif images that were generated upon segmentation of the reconstructed volume by replacing the value of the pixels outside of the sample by zero.Azimuthal integration.The azimuthal integration generated the gray value maps showing the evolution of the gray values along the gel radius and height over time (GHR maps).The procedure was performed on the masked slices using the Python library DipLib [1].The center of the sample cross-section in the image was calculated as the center of mass (first order moments) of the image with the function diplib.CenterOfMass().The azimuthal integration was then computed with the function diplib.RadialMean() using a bin size of one pixel.Integrated gray values at a radial distance from the center larger than 1.05 times the average radius of the gel at a given scan number were set at zero.This was done to limit the propagation of the imperfect segmentation at the bottom of the gel.Additionally, if the area of the sample in the image was less than half of the mean cross-section area of the gel, the integrated gray value of the masked slice was set to zero.This criterion allowed to exclude the noise at the top of the reconstructed volume due to imperfect segmentation.The mean cross-section area was calculated from the average diameter of the gel from our recent work [2].
The profiles created by azimuthal integration of all masked slices at a given scan number  were then combined into a single map, referred to as  � ,, , where  ∈ ℕ represents the radial distance to the center of the cylinder in pixels.The gray values were then corrected for the anode heel effect with: where  ,, are the maps corrected for the anode heel effect,   is the pivot slice number and  * () is an exponential decay function.The parameters used in eq. ( 1) were the same as in ref. [2] (see Supporting Information, section SI3).The GHR maps were saved as npy files for further processing and as 8-bit tif images for illustration purposes.Figure S2 illustrates the azimuthal integration procedure.Azimuthal and vertical integration.The azimuthal and vertical integration generated a single radial gray value map (GR map) for a given sample, showing the evolution of the gray values along the gel radius over time.The GHR maps generated by azimuthal integration were further integrated along the index  (along the height of the gel) to compute the GR map: where  , is the GR map,  Ω , is the height in pixel of the domain Ω , which defines the range of  indexes belonging to the sample.Note that the domains Ω , were cropped, so that the sum in eq. ( 2) was effectively done by excluding a top and bottom band of 70 pixels.This was done as an attempt to get a more representative evolution of the gray values along the gel radius over time.Moreover, since the GHR maps were already corrected for the anode heel effect, no additional correction procedure was required.
Slice integration.The slice integration generated a single vertical gray value map (GH map) for a given sample, showing the evolution of the gray values along the gel height over time.In practice, this reduction procedure was carried out directly in the software Dragonfly [3] using the "slice analysis" plugin.The masked slices were integrated along the indexes  and  as follows: where  � , is the uncorrected GH map,  Ω , is the area in the masked slice within the domain Ω , defining the range of  and  indexes belonging to the sample.The map  � , was then corrected for the anode heel effect with a similar expression as in eq. ( 1).

SI2: Derivation of the µCT Drying Model
Here the µCT drying model is derived by developing the equations step-by-step, leading to the final expression for the spatial and temporal volume fraction maps of the three phases composing the gel.As mentioned in the main text, the reconstructed attenuation coefficient (RAC) of each pixel belonging to the sample in the MHR maps is given by: where  hex and  skel is the RAC of the hexane and skeleton phases, respectively and  hex,,, and  skel,,, are the volume fraction of hexane and skeleton in each pixel. hex = 0.155 from separate measurements on hexane.Volume conservation within each pixel of the MHR maps reads: hex,,, +  skel,,, +  air,,, = 1.
Eqs. ( 4) and ( 5) were the main equations of the drying model.To calculate the volume fraction maps of each phase, some assumptions were made on the evaporative drying process.First, the total skeleton volume was assumed constant throughout drying:  skel, = constant, which can be expressed locally as: �  skel,,,  voxel Second, the content of hexane after a certain time was assumed to be zero everywhere in the gel, leading to:  hex,,,≥  = 0, with   a threshold scan number upon which the gel was assumed dry.Lastly, the initial content of vapor/air was assumed to be zero everywhere in the gel, leading to:  air,,, 1 ≤≤ 2 = 0 where  1 and  2 are threshold scan numbers.Scan  1 defined the time from which the instabilities in the X-ray tube disappeared and scan  2 stood for the time where vapor/air started to enter the gel.  ,  1 and  2 were calculated using the global quantitative imaging approach reported in ref. [2].Additionally, a variation of the skeleton volume conservation equation can be derived by taking the sum over the domain Ω  of eq. ( 4): �  ,, −  hex  hex,,, −  skel  skel,,, ⇔ �  ,, −  hex  hex,,, Where eq. ( 8) was derived using the conservation of the skeleton volume.Taking eq. ( 5) for  1 ≤  ≤  2 gives: hex,,, = 1 −  skel,,, ,  1 ≤  ≤  2 . (9) By replacing  hex,,, in eq. ( 8) by its expression in eq. ( 9), we get: Eq. ( 11) was not an additional equation per se, but will be used later on.

Hexane volume fraction.
We first derive the expressions used to calculate the hexane volume fraction maps throughout drying.To improve the statistics, the calculations were done on a MHR map representative of the dry gel for  ≥   .The hexane volume fraction  hex,,, is calculated by finding an expression for  skel  skel,,, in eq. ( 4).For  ≥   , eq. ( 4) becomes: because there was no more hexane in the gel at  ≥   .By combining the MHR maps  ,, over  ≥   , an artificial MHR map representative of the dry gel could be generated.However, the volume of the gel still changed at  ≥   , so did the domains Ω  , preventing to average the MHR maps directly.We write formally: This was solved by resizing one map onto another using bilinear interpolation so that the domain of the maps matched.The MHR map to be resized was referred to as the source and the map over which it was resized was referred to as the target.In this case, the target was unique and was set as the MHR map of the last scan ( =   ), while the source scans were multiple (  ≤  <   ).Note that the target scan could have been any scan within  ≥   .We define: The algorithm for the bilinear interpolation function is described in SI3.A series of rescaled maps  ,,→  * for   ≤  <   was thus obtained, each map being defined on the same domain Ω   .To derive the scaling factor in eq. ( 14), let's consider a quantity that stays constant throughout drying.We recall the conservation of the skeleton volume: �  skel,,,  voxel By substituting  skel,,, in eq. ( 6) by its expression from eq. ( 12), we get: where ̅  is the RAC averaged over the domain Ω  .Since the average of all values of a MHR map (or any digital image) are conserved upon interpolation, we have that: giving an expression for the scaling factor : The expression of the  scaling factor can be generalized: The rescaled maps were averaged over the scans  ≥   , resulting in an artificial MHR map representative of the dry gel: where   is the number of scans in   ≤  ≤   .The quantity  ,,  dry was referred to as the dry MHR map.
By recalling eq. ( 12), the MHR maps  ,,≥  were equal to  skel  skel,,,≤  under the assumption of a zero hexane content in the gels at  ≥   .The dry MHR map derived in eq. ( 24) was thus an expression of the quantity  skel  skel,,, interpolated onto scan  =   .We set: The dry MHR map was then used to calculate the HEXHR maps.Let's recall the expression of the local RAC: At  =   , the rightmost term in eq. ( 4) is equal to the dry MHR map.The change of the gel volume throughout drying implied that Ω ≠  ≠ Ω   .By assuming that the distribution of the silica skeleton within the gel's volume does not change throughout drying, an expression for  skel  skel,,, for any scan  could be obtained by rescaling  skel  skel,,,  (which is known) from source scan  =   to target scan  ≠   using a similar methodology as before.In this case, the source is unique ( =   ) and the targets are multiple ( ≠   ).We thus set: where  skel  skel,,, * is the rescaled map and the scaling factor is: The expression for the scaling factor was verified by recalling the conservation of the skeleton volume, considering the rescaled map at scan  ≠   and the dry MHR map at scan  =   : where ̅   → dry, and ̅   dry is the RAC of the maps averaged over the domains Ω  and Ω   , respectively.Finally, the quantity  skel  skel,,, in eq. ( 4) was replaced by the expression in eq. ( 28), giving an expression for the hexane volume fraction maps at any scan: Skeleton volume fraction.The SKELHR maps can be directly calculated at  1 ≤  ≤  2 using the volume conservation equation and the previously computed HEXHR maps.Similarly to the procedure adopted to calculate the hexane volume fraction maps, an artificial MHR map representative of the state of the gel at the beginning of drying was computed to improve the statistics upon calculating  skel,,, .This was done by rescaling and combining the MHR maps over  1 ≤  ≤  2 .The rescaling was performed from source scans  1 <  ≤  1 onto target scan  =  1 .In this case, the target is unique and the sources are multiple.The rescaled maps are: where  ,,→ 1

𝐹𝐹
is the interpolated map defined over the domain Ω  1 and  is an additional scaling factor.An expression for  was found by considering a quantity that stays constant over  1 ≤  ≤  2 (similar strategy as the one employed to determine ).Let's recall the 2 nd conservation equation: Let's consider eq. ( 11) between a rescaled MHR map from scan  ≠  1 to  =  1 and the MHR map of source scan : where ̅ → 1

𝐹𝐹
and ̅  are the MHR maps averaged over their domain Ω  1 and Ω  , respectively.Because bilinear interpolation conserves the average value of a MHR map, we have that: giving an expression for : The rescaled MHR maps were then averaged over  1 ≤  ≤  2 , resulting in an artificial MHR map representative of the state of the alcogel referred to as the alco MHR map: where   is the number of scans between  1 ≤  ≤ The SKELHR maps at scans  ≠  1 were determined by rescaling  skel,,, 1 : where  skel,,, 1 →  is the interpolated map defined over the domain Ω  .
Vapor/air volume fraction.The vapor/air volume fraction maps were directly computed using eq.( 5) with the knowledge of  hex,,, and  skel,,, :

SI3: Bilinear Interpolation Procedure
A key step in the local quantitative imaging procedure was the resizing of the maps between the different drying stages by bilinear interpolation.The resizing and rescaling of a GHR map from source scan   = 140 onto target scan   = 0 is illustrated in Figure S3 as an example and consisted in: (i) conversion of the GHR maps into MHR maps (Figure S3b), (ii) define the edges of the domains Ω   and Ω   separating the sample from the background in the source and target MHR maps (Figure S3c), (iii) compute the normalized vertical and radial coordinates of any pixel of the maps within the domain edges (not shown), (iv) bilinear interpolation of source map onto target map and correction of the RAC values by a scalar factor (Figure S3d).As mentioned in the main text and in SI2, the domain of two GHR or MHR maps taken at different scans did not match, due to the shape change of the sample.To compare maps at different drying stages required establishing a correspondence between the two domains, which was done by bilinear interpolation of a source scan:   towards the domain of a target scan:   .The procedure described here is based on the GHR maps but can also be applied to a MHR map or a volume fraction map upon minor adaptations.First, the edges of the domains were defined as the limit where the gray values dropped to zero in the GHR maps (Figure S3c).The top edge of the gel was called the north edge:  , , and was defined as the  index where the gray values became non-zero, from top to bottom for each index .The radial edge of the gel was called the east edge:  , , and was similarly defined as the  index where the gray values became non-zero, from right to left for each index .The bottom edge of the gel was called the south edge:  and was independent of the radial position and of the scan number.Figure S4 depicts those edges of the GHR maps at scans   and   shown in Figure S3.We define the normalize coordinates within a domain Ω  as: 1) Define the vertical and radial tolerance: where    min is the minimum of  ,  and    max is the maximum of  ,  .
2) Set a starting point in the source map (ℎ 0 ,  0 ) as: where the notation ⌊⌉ stands for rounding  to the nearest integer (0.5 is rounded down to 0).
3) Compute the normalized radial coordinate in the source map  1 the closest to the target coordinate  * at ℎ = ℎ 0 : The function argmin is the argument of the minima taken over a series of coordinates (, ), where  is such that ℎ , = ℎ 0 .This represents a relatively horizontal line.
4) Do the same for the normalized vertical coordinate ℎ 1 at  =  1 : where the coordinates  is such that  , =  1 .
Once the correspondence between the normalized coordinates in the source map (ℎ, ) and in the target map (ℎ * ,  * ) was established, an empty map with the domain of the target map Ω   was created.The gray values in the empty map were filled using a bilinear interpolation algorithm.For any pixel with the coordinates ( * ,  * ) in the empty map, its gray value was determined by: 1) Calculate the virtual position in the source map corresponding to the normalized coordinates (ℎ * ,  * ) of the empty map: where  and  are not pixel indexes but a virtual position in between pixels.(, ) are known from �ℎ , ,  , � determined in the previous calculations.
2) Find the four pixels the closest to the virtual position (, ): 3) The gray value in the empty map at the coordinates ( * ,  * ) is given by the bilinear interpolation formula: By construction, the interpolated map   * , * ,  →  was defined over the same domain as the target map  ,,  .
After interpolation, depending on the nature of the interpolated map (GHR, MHR or volume fraction map), the interpolated map was corrected with one or more scaling factor (see SI2).

SI4: Gel Diameter during X-ray scattering measurements
Here we describe how the local diameter of the gels was calculated during their drying at the µSpot beamline at BESSY.The digital pictures were correlated with the µCT data and with the time stamps of the scattering data.Figure S6 shows a series of digital pictures at selected drying stages, where the maximum shrinkage occurred after ca.4.4 h of drying, which was faster than in the µCT experiments.Due to the insufficient contrast in the image, the dimensions of the gel could not be retrieved by automated image processing.Instead, the height of the gel in pixel was measured manually from 14 pictures using the software Fiji [4].It must be noted that during the first 2 h of drying, the height of the gel could not be measured because its top was hidden by the measurement cell (Figure S6).The diameter of the gel in pixel was derived by dividing the height by a factor 2 (gels have an aspect ratio of 2 [2]) and was normalized by the diameter of the gel at the maximum shrinkage.We define: with  PI ( PI ) the normalized diameter from the digital pictures,  PI ( PI ) the diameter of the gel in pixels and  PI,MA the time of maximum shrinkage. PI is the time at which the digital pictures were recorded. PI ( PI ) is shown in Figure S7.Because the diameter of the gel at  PI < 2 h could not be measured, the diameter from the digital pictures was fitted using µCT data, allowing to extrapolate the diameter of the gel at the start of drying.
During the in operando X-ray scattering experiment, the X-ray beam probed the gel at a vertical height of ca. 4 mm from the bottom of the gel.That location can be seen in Figure S6 as a slight darker line on the museum glass.The diameter of the gels dried by µCT at the same location was extracted from the masked images and was normalized by its value at the maximum shrinkage as in eq. ( 64).The corresponding normalized diameters of samples M1, M2, M4 and M5 are shown in Figure S7.Note that the drying rate within these samples differed because of slightly different starting volume of the gels and possibly drying conditions.The µCT normalized diameter was taken as the average among samples M1, M2, M3 and M4 to improve the accuracy of the fit.To do so, the µCT time scale was also normalized: where  CT, is a time scale normalized over the time of the maximum shrinkage  CT,MS, for a sample .
The average µCT normalized diameter was then:  CT ( CT ), where  CT = 0 and  CT = 1 corresponded to the start of drying and to the maximum shrinkage, respectively.
The normalized diameter from the digital pictures was also expressed as a function of a normalized time scale.However, the first recorded image was taken a few minutes after the gel started drying due to experimental limitations.We defined the time difference between the first recorded image and the effective start of drying of the gel as Δ 1 .Therefore, we can express  PI =  PI ( PI ) with  PI as: By minimizing the difference of � CT ( CT ) −  PI ( PI )� 2 , Δ 1 could be determined at ca. 3.5 min, which was reasonable.After a time scale normalization,  PI ( PI ) was in good agreement with the normalized diameter from the µCT data (see Figure S8). PI ( PI ) was then interpolated at the points  CT .The last step was to interpolate the diameter of the gel with the time of the scattering data.The scattering intensity could be expressed as (,  XS ) with: where  XS is the normalized time scale of the X-ray scattering data frames,  XS the time in h,  XS,MS the time of maximum shrinkage and Δ 2 a time shift. XS,MS was set by visual inspection of the scattering profiles (see Figure S19).Δ 2 was known, it corresponded to the delay between the first scattering data frame recorded and the first digital pictures that was taken. PI ( PI ) was then interpolated at the points  XS .The diameter in absolute units was calculated with:  XS ( XS ) =  PI ( XS ) ⋅  CT,MS , from which the diameter of the gel during drying was found by converting  XS back to the absolute time scale  XS .

SI5: Spatial Variability Analysis
Vertical and radial distributions.This section reports the spatial variability analysis of the gray values in the GHR and GR maps and discusses their origin.A heterogeneous distribution of the gray values was observed in the GHR maps (Figure 3, Figure S9 -Figure S11).To further quantify these variations, gray value profiles were extracted from the GHR maps of sample M4 across the gel height and radius as shown in Figure S22.Three vertical profiles were extracted at a relative radius of 0.25, 0.5 and 0.75 (from the center of the gel to its radial edge) and three radial profiles were extracted at a relative height of 0.25, 0.5 and 0.75 (from the bottom to the top of the gel).The gray values remained relatively constant across the gel's height throughout drying, suggesting a homogeneous composition (Figure S22a-c).Variations were only observed during the spring-back effect (visible in Figure S22c at 7 -8 mm and 7.6 h of drying) and at the edges of the sample, regardless of the radial coordinate.The small peak at a height of ca.0.5 mm was seemingly caused by under-sampling artifacts in the masked slices [5] due to the low number of projections taken and the proximity of the sample with the drying chamber.This region with locally higher gray values can clearly be seen in the GHR maps (Figure S9 -Figure S11).The lower gray values before the peak at the bottom of the gel (at a height of 0 mm), as well as the lower gray values at the top of the gel (Figure S22a-c) were due to imperfect segmentation at the edges which included some of the background in the ROIs.
Across the gel's radius, the gray values followed an exponential growth with the radius regardless of the vertical coordinate (Figure S22d-f).After reaching a maximum near the edge of the sample, the gray values decreased due to imperfect segmentation that included some of the background in the masked slices.The exponential dependency of the gray values on the gel's radius was coherent with beam hardening artifacts in cylindrical samples [6].Beam hardening artifacts are caused by a higher attenuation of the soft X-rays in the center of the sample than at its periphery with polychromatic radiations.Beam hardening was expected in our in operando experiments given the relatively high voltage (135 kV) and the absence of a filter.The change in amplitude and curvature of the radial variations was quantified throughout drying by fitting the gray value profiles to compare the variations with a simplified beam hardening model.RAC radial profiles were extracted from the MR map of sample M4 at each scan number.The edges effect were excluded by considering the profiles until the maximum of the RAC values.The RAC radial profiles were then fitted with an exponential function defined as () =    ⁄ + , where r is the gel's radius in mm and A, B and C are fitting parameters.Figure S23a shows the fitted radial profiles at selected drying stages.
The amplitude of the radial variations was quantified by   , defined as the difference of the RAC at the outer radius of the gel and the RAC at the center of the gel: where  max is the maximum radius of the gel.The curvature was quantified with the inverse of the  parameter, the lower  the more the profiles were curved, compared to flat profiles as  → ∞.The fitting of the radial profiles generated 141 values of   and  shown in Figure S23b and Figure S23c, respectively along the average diameter of the gel.  was inversely proportional to the gel diameter: the amplitude of the radial variations increased as the gel diameter decreased.The maximum amplitude corresponded to the maximum shrinkage and the steep drop afterwards was likely caused by the heterogeneous springback of the gel.A similar behavior was observed for the curvature of the spatial variations.The evolution of the  parameter throughout drying was comparable to the gel diameter with a higher curvature the smaller the diameter was.Identical conclusions were made from spatial variability analyses on samples M1, M2 and M4.This procedure was also performed on the hr-maps and provided similar results in terms of amplitude, but the curvature could not be accurately determined due to the noise in the GHR maps.The correlation between the amplitude (  ) and the curvature (1  ⁄ ) of the variations with the diameter of the gel was consistent with beam hardening.Considering a large cylindrical sample, most of the soft X-rays are absorbed except a relatively small portion at the sample edges resulting in virtually higher RAC in these regions.In a smaller cylindrical sample of the same material, the difference in the extent of soft X-rays absorbed in the center and at the edge is more pronounced due to the exponential dependency of X-ray transmission with the sample thickness 1 [7], resulting in stronger beam hardening and in increased amplitude and curvature of the variations in the gray values and RAC along the gel radius.
Besides beam hardening artifacts, a potential gradient of the hexane composition along the radial direction of the gel could also generate the radial variations observed in Figure S22d-f and Figure S23a.However, the amplitude of the variations would then vanish in the dry gel, which was not the case in our experiments.
Additionally, the value of   was inversely proportional to the diameter of the gel and not on the hexane content.Eventual radial variations caused by a different hexane content should decrease in amplitude as 1 The transmission of a monochromatic X-ray beam through a materials of attenuation coefficient  and thickness  is given by Beer-Lambert's law:  =   0 ⁄ =  − , where  is the transmission and  and  0 are the transmitted and initial intensity of the X-ray beam, respectively.Azimuthal variations.This section presents the spatial variability analysis on the gray values along the azimuth of the cylindrical samples to evaluate potential heterogeneities in the gel composition during drying.
To do so, selected masked slices were integrated radially with a custom Python script to produce azimuthal gray value profiles.For each scan number, a masked slice was selected at a given relative height of the cylinder.The center of each masked slice was defined as the center of mass as described in SI1.The pixels in the masked slices were separated into 100 bins depending on their azimuth from 0 to 2 and were radially integrated.The resulting gray value maps were defined as the azimuthal maps  , , where  stands for the azimuthal range with 0 ≤  ≤ 99 and  for the scan number.Rather than evaluating the azimuthal map, the difference of the gray values compared to the average at a given scan was computed: where  , is also a map. Figure S25 shows the final results at three relative heights: 0.25, 0.5 and 0.75.Slight variations in the gray values (+/-1 gray value) were consistently observed along the azimuth regardless of the drying stage and relative height.The gray values were higher than the average between  2 ⁄ and  and lower between 3 2 ⁄ and 2.Similar variations were observed in the other samples at different angles, and the minimum of the variations systematically occurred in the regions the closest to the wall of the drying chamber while the maximum occurred at the opposite location (Figure S26).Those spatial variations were attributed to reconstruction artifacts but could not be clearly identified to a specific µCT artifact.A combination of undersampling and beam hardening artifacts may have created these variations.
Nevertheless, the established dependency of the azimuthal variations with the proximity of the gel to the chamber suggested that they did not originate from heterogeneities in the composition of the gels.Stronger azimuthal variations appeared shortly after the maximum shrinkage in samples M2, M4 and M5 (Figure S25a,b and Figure S26).The time at which the variations increased corresponded to the time where the drying front reached the relative height of the gel at which the azimuthal maps were produced, which suggested that the spring-back effect was slightly heterogeneous along the azimuthal direction of the gels.This feature was not observed in sample M1, which may be related to the fact that M1 stayed relatively well centered during drying.Sample M5 moved during drying, resulting in a change of direction of the variations before and after maximum shrinkage (Figure S26d).

SI6: Comparative analysis of the Quantitative Imaging Approaches
This section reports a comparative analysis between the quantitative imaging results from this study and from the global quantitative imaging procedure from ref [2] to evaluate the reliability of the approach presented here.The gray value and volume fraction maps from the three reduction procedures were integrated along their corresponding spatial domains to calculate a spatially averaged quantity representative of the gel throughout drying.The integration of the volume fraction maps was performed over a partial domain of the corresponding maps and was consistent with the global quantitative imaging approach.For a given map  ,, , the integration was defined as: Where  �  is the average value of the map  ,, integrated over the partial domain Ω  ′ , with  ,, a GHR, HEXHR, SKELHR or AIRHR map,  Ω  ′ is the number of pixels within Ω  ′ and   is a radial weight factor depending on the radial distance given by the index .As a reminder to the reader: the GHR maps were

Figure S1 .
Figure S1.Sketch of three reconstructed volumes along with the three spatial axes where the indexes , ,  are defined and the time axis where the index  is defined.

Figure S2 .
Figure S2.Illustration of the reduction procedure by azimuthal integration on sample M2 at the start of drying.(a) µCT projection at the start of drying.(b) Reconstructed slice at  = .(c) Same slice overlaid with the ROI from the automated segmentation.(d) Corresponding masked slice.(e) Sketch of the azimuthal integration on the masked slice.(f) Final GHR map created by combining the radial gray value profiles generated by the azimuthal integration at each index .The scale bar in each panel is 5 mm.The gray values are in 8-bit.The arrow in each panel stands for the vertical direction of the gel.

Figure S3 .
Figure S3.Example of the bilinear interpolation procedure and rescaling.(a) GHR maps of sample M4 at 10 selected drying stages on top of a cyan background with the corresponding color scale on the right.The brightness and contrast in the images of the gray value maps was adjusted to improve visualization.(b) MHR maps converted from the GHR maps at the start of drying (target scan) and at the end of drying (source scan) (c) Domains defining the sample in the MHR maps (white line over black background) in target and source scans.The red squares illustrate the correspondence in the relative coordinates of two pixels in both scans.(d) Target MHR map and rescaled MHR map interpolated from the source scan domain onto the target scan domain.The color scale of the MHR maps in panels (b) and (d) is shown at the bottom right of the figure.The length axes of all maps is indicated in the first map of panel (b).

Figure S4 .
Figure S4.Edges (white lines) of the domains of two GHR maps from (a) a target scan at the start of drying (k=0) and (b) a source scan at the end of drying (k=140).The north edges are depicted with an arrow:  ,  and  ,  for target and source scans, respectively.The east edges are depicted accordingly:  ,  and  ,  .The south edge  is also shown and is independent of the  index and of the scan number .The axes are shown in red on the top left of the figure in panel (a).

Figure S5 Figure
Figure S5

Figure S6 .
Figure S6.Digital pictures of the gel dried in the measurement cell during the in operando SAXS/WAXS measurement.The picture corresponding to the maximum shrinkage is outlined in red.The bright spots are due to the reflection of the camera lamp on the museum glass.The brightness and contrast of the images are adjusted for better visualization.

Figure S7 .
Figure S7.Diameter of a gel normalized over its diameter at the maximum shrinkage:  from the µCT measurements (M1, M2, M4 and M5) and from the digital pictures versus the drying time.

Figure S8 .
Figure S8.Normalized diameter  from the µCT and digital pictures expressed as a function of a normalized time scale.

Figure S9 .
Figure S9.GHR and volume fraction maps of sample M1 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 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 1.64 +/-0.04 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 in 410 x 1455 non-interpolated data points.

Figure S10 Figure
Figure S10

Figure S11 Figure
Figure S11

Figure S14 .
Figure S14.Radial and vertical maps of the gray values and volume fraction of sample M5 on top of a cyan background.The 3D image on the left of the figure depicts the segmented volume of M5 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 in 1444 x 410 data points and the vertical maps in 1444 x 1455 points.In all maps, the horizontal time resolution is interpolated from 141 time stamps onto 1444 points.

Figure S20 Figure
Figure S20

Figure S22 .
Figure S22.Gray value profiles extracted from the GHR maps of sample M4.Each panel presents the gray value profiles at six drying stages.Each panel includes a GHR map as an inset with an arrow illustrating the region where the gray value profiles are extracted.The regions (lines) are fixed at a relative height or relative radius regarding to the maximum height or radius at each drying stage.(a-c) Gray value profiles along the height of the gel at a relative radius of (a) 0.25, (b) 0.5 and (c) 0.75.(d-f) Gray value profiles along the radius of the gel at a relative height of (d) 0.75, (e) 0.5 and (f) 0.25.

Figure S23 .
Figure S23.(a) Example of the fitting results on the radial profiles extracted from the MR maps of sample M4 at six drying stages.(b) Amplitude of the gray value variations:   parameter (blue) and average diameter of the gel (orange).(c) Curvature of the gray value variations (blue) and average diameter of the gel (orange).The smaller , the more curved the radial profiles.

Figure S24 .
Figure S24.Hexane, skeleton and vapor/air radial profiles extracted from the HEXR, SKELR and AIRR maps of sample M4 at three selected drying stages: (a) 0.0 h, (b) 6.0 h and (c) 14.0 h.The time is indicated in each panel on the top right corner.

Figure S25 .
Figure S25. , maps (gray value difference) of sample M4 throughout drying.The maps in panels (a-c) refer to the radial integration performed at a relative height of (a) 0.25 (near the bottom), (b) 0.5 and (c) 0.75 (near the top) in the gels.Each panel share the same color scale which is indicated at the bottom right of the figure.Each map consists in 100x141 data points and the horizontal axis is linearly interpolated in time.

Figure S26 .
Figure S26. , maps (gray value difference) at a relative height of 0.5 throughout drying.The maps in panels (a), (b) (c) and (d) refer to the samples M1, M2, M4 and M5, respectively.Each panel contains an inset on the right of the map showing the masked slice at the relative height of 0.5 at the beginning of drying.The angle corresponding to the maximum (yellow) and minimum (blue) of  , are depicted on the masked slice with arrows.Each map consists in 100x141 data points and the horizontal axis was linearly interpolated in time.

Figure S27 .
Figure S27.Comparison between the local and global quantitative imaging approaches in sample M1.(a) Global gray values and averaged local gray value in the three reduction procedures.(b-d) Global volume fraction (dashed lines) and averaged local volume fraction (full lines) of the three phases for the azimuthal integration (b), slice integration (c) and azimuthal + vertical integration (d) reduction procedures.

Figure S28 .
Figure S28.Comparison between the local and global quantitative imaging approaches in sample M2.(a) Global gray values and averaged local gray value in the three reduction procedures.(b-d) Global volume fraction (dashed lines) and averaged local volume fraction (full lines) of the three phases for the azimuthal integration (b), slice integration (c) and azimuthal + vertical integration (d) reduction procedures.

Figure S29 .
Figure S29.Comparison between the local and global quantitative imaging approaches in sample M4.(a) Global gray values and averaged local gray value in the three reduction procedures.(b-d) Global volume fraction (dashed lines) and averaged local volume fraction (full lines) of the three phases for the azimuthal integration (b), slice integration (c) and azimuthal + vertical integration (d) reduction procedures.

Figure S30 .
Figure S30.Comparison between the local and global quantitative imaging approaches in sample M5.(a) Global gray values and averaged local gray value in the three reduction procedures.(b-d) Global volume fraction (dashed lines) and averaged local volume fraction (full lines) of the three phases for the azimuthal integration (b), slice integration (c) and azimuthal + vertical integration (d) reduction procedures.

Figure S31 Figure S31 .
Figure S31 and  is a correction factor.Formally, we also define  →  as the bilinear interpolation function from a source  towards a target   : skel,,,  → * =    → � skel  skel,,,  � ⋅    → (26) ⇔  skel  skel,,,  → * =    → � ,,  dry � ⋅    → , (27) ⇔  skel  skel,,,  → * map representative of the state of the alcogel expressed at scan  =  1 .This expression was finally used to calculate the SKELHR map at scan  =  1 from the conservation of the total 2 .The alco MHR map was used to compute a hexane volume fraction map representative of the alcogel at scan  =  1 .Eq. (33) was rewritten considering a target scan  =  1 and by replacing the MHR map at scan  by the alco MHR map at scan  =  1 : alco is a hexane alco .