A facile way of controlling capillary condensation: particle-based crystal

To demonstrate a facile method to control capillary condensation via the design of confined geometric structures, we use a particle self-assembly technique to fabricate porous materials with well-defined pore sizes. Four groups of silica particles were synthesized using the modified Stober method, and these groups of particles were then arranged in closely packed structures. Quantitative predictions of capillary condensation were made based on the Kelvin equation and an approximation of the geometric structures of our closely packed samples. Experimental observations revealed that water uptake at 100% relative humidity reached 40%–50% relative to particle mass across sizes, closely aligning with theoretical predictions for small-particle systems, even though the geometry of some of the confined spaces corresponds to distances smaller than 10 nm. Despite deviation from theoretical predictions that are observed in larger particle systems, and which can be attributed primarily to practical limitations of attainable ordered structures at these scales, this fabrication method shows great potential for the creation of devices that allow facile control of capillary condensation in relevant applications such as vapor capture and humidity control.


Introduction
It is well known that a confined fluid's phase behavior differs from that of the bulk [1][2][3][4].In particular, under nanoscale confinement, the observed phase behavior is influenced by liquid-wall/pore interactions.Capillary condensation is a resulting phenomenon that significantly impacts many fields relevant to water [5,6], air [7], and energy [8][9][10][11][12].While a continuum-level understanding has been well-established, the initiation of nucleation and the propagation of capillary condensation in nanostructured materials is not fully understood.In recent years, both theoretical and experimental studies have been aimed at establishing a better understanding of capillary condensation, especially down to the sub-10 nm scale [8,13,14].
The foundation of continuum-level theoretical studies of phase behavior under confinement is the centuryold Kelvin equation [15] derived from the Young-Laplace equation [16].The Kelvin equation has been used extensively since its development to describe capillary condensation [14,[17][18][19][20][21], primarily at scales where the continuum approximation is well accepted.On the other hand, there has been some recent debate regarding the validity of the Kelvin equation down to the sub-10 nm level [22][23][24].Thus, in the past two decades, researchers have examined alternative ways to describe this phenomenon, including the development of models that consider the thermodynamic energies throughout all phases using such techniques as density functional theory [1,15,25,26], grand canonical Monte Carlo simulation [27,28], and Molecular Dynamics [29,30].
At the same time, experimental studies have aimed to validate these theories and models as well as to provide new insight into existing theories.Needless to say, directly investigating the nucleation site at the single molecular level is challenging.Thus, the majority of the prior experimental studies involve interpreting the absorptive data from standard porous media, including hydrophilic solids such as silica gels, zeolites, and some classes of metal-organic frameworks [8,13].More recently, technological advances have enabled researchers to probe capillary condensation at the sub-10 nm level via a number of techniques.For example, measurement of a 'critical' tip-surface distance can be made using atomic force microscopy (AFM) to identify when capillary condensation/nucleation occurs.In order to accomplish this measurement, the sharp AFM tip is slowly made to approach a surface.When a sudden drop of the tip's amplitude is realized, it is inferred that the spontaneous formation of a water meniscus has occurred, and the 'critical' distance is realized (and interpreted to represent the distance necessary to induce nucleation at a particular relative humidity).By varying the relative humidity in the environment of this apparatus, one can obtain a series of experimental values of the critical distance corresponding to different relative humidities [14].While this technique is quite powerful, it is difficult to experimentally maintain both ultra-low (i.e., 2%) and ultra-precise humidities for the time required to perform the AFM experiments.This may explain why, for very low humidities, the measured critical distance is found to be significantly different from the theoretical values that are predicted by the Kelvin equation.In contrast, several competing experimental efforts have appeared to support the validity of the Kelvin equation at the sub-10 nm scale [20,[31][32][33].For example, one group used a nanofluidic device [20] to help understand capillary condensation at the nanoscale.They directly measured condensation initiation and dynamics within 8 nm deep silicon nanochannels.Their results suggest that the initiation of capillary condensation in the sub-10 nm scale agrees closely with the Kelvin equation.Similarly, a group recently studied condensation using atomic-scale capillaries created by two-dimensional crystals, forming capillary geometries as small as four angstroms.The observations in that work suggested that the Kelvin equation can quantitatively describe capillary condensation at scales even smaller than a nanometer [21].
As a complement to these detailed experimental studies, this paper demonstrates a facile method to manipulate capillary condensation over a range of humidities by placing particles onto surfaces in order to create confined geometric structures.Specifically, highly controllable and ordered porous structures are created using a simple particle self-assembling technique.Using these surfaces, we are able to demonstrate the ability to control the capillary condensation behavior by changing the size distribution of the particles that are deposited on the surfaces.Theoretical predictions based on the Kelvin equation show agreement with adsorption experiments that not only demonstrate that this methodology is effective at condensation control, but these results also lend support to the applicability of the Kelvin equation at scales below that of 10 nm.We believe that further exploration of this method could lead to exciting applications relevant to capillary condensation, such as vapor capture, humidity control, and more.

Experimental methods
In this section, we describe the materials and method to synthesize silica particles and the preparation of the multilayer structure.The characterization methods and equipment settings will also be introduced.

Materials and characterization
Tetraethyl orthosilicate (TEOS), ethanol (C2H5OH), and ammonium hydroxide solution (NH4OH) were obtained from Sigma-Aldrich.The silicon wafers used were obtained from University Wafer Inc.Where applicable, the particle size distribution was characterized by Zetasizer Nano S. This was accomplished as follows.The particle suspension was diluted prior to the measurement.1 ml of the suspension solution was added to the disposable cuvette, which was then inserted into the instrument.At least three independent runs were taken for each sample, and the count rate and correlation factor were checked to ensure the quality of the measurements.The structure of the particle layers on the silicon wafer was characterized using the scanning electron microscope (SEM) Zeiss Sigma 500 VP Analytical FE-SEM.The operation voltage varied from 1 kV to 10 kV depending on the particle size and the layer thickness.The morphological structure of the sample was also characterized by an atomic force microscope (AFM) using a Bruker Nano Multimode.

Preparation of silica particles
Silica particles with different size distributions were synthesized using the modified Stober method under room temperature conditions [34] The appropriate amount of ethanol, deionized water, and ammonia solution was mixed in a 20 ml transparent glass vial.Then a certain quantity of tetraethyl orthosilicate (TEOS) solution was added according to the desired ratio (appropriate to the target particle size) in the mixed solution.The vial was vigorously shaken for 30 s, and then the reaction was allowed to proceed at ambient temperature for 12 h.The solution can be used as-is or diluted for further analysis and experiments.Particles of 4 different sizes are successfully synthesized using the modified Stober method (see figure 1 for the relevant size distributions).All the particles prepared are stable, and the size distribution is relatively narrow as the curve is single-peaked with extended tails.Furthermore, the polydispersity is similar across all the experiments.Several factors contribute to polydispersity.Generally, the Stober method consists of two major mechanisms: first, TEOS polymerizes to form nano particles, so called primary particles; Second, as the primary particles increase in size, they can come into contact and aggregate with each other to form larger particles.Varying balances between the rates of nucleation, growth, and aggregation can lead to polydispersity.For example, under certain conditions, the rate of nucleation of new primary particles exceeds the rate of consumption of reagents for the growth of existing particles.This leads to an abundance of new primary particles and a smaller mean particle size.Alternatively, if the rate of growth and/or aggregation is faster than that of nucleation, the synthesis will lead to particles of larger mean size.When each of these rates are comparable, the synthesis leads to polydispersity [35].

Preparation of the multilayer structure
The multilayer closely packed particle matrix was prepared via the evaporation self-assembly process.After synthesis, the particle suspension was prepared by dilution of the as-is solution with ethanol at a ratio of 1:4.The silicon wafer was cut into desired sizes and the diluted suspension was then deposited, via pipette, onto the precut clean silicon wafer surface and let dry in the air.Depending on the experimental conditions, a heating plate or vacuum chamber may be used to speed up the evaporation process.
In the results quoted here, the self-assembled particle bed is created through the evaporation of the solvent (i.e., ethanol in this case) that results in capillary forces arising between particles as the shrinking liquid volume forms inter-particle liquid bridges [36].Figure 2 shows an AFM image of 35 nm particles deposited on a silicon wafer as described.The image shows that the spherical particles are agglomerated together, forming a multilayer structure.Figure 3 shows an SEM image of the 200 nm particle sample's deposition results.It can be seen that even for the largest particles used in our experiments, a closely packed structure is achieved via the fast evaporation method.Moreover, figure 3(b) shows that the particles tend to aggregate with others of similar sizes and particles with smaller sizes tend to accumulate near the edge of the particle blocks.One can also notice defects in the multilayer structures due to the bubble formation inherent in the fast evaporation process.Since the defects are much larger in relative terms to the particles, the resulting pore spaces do not contribute significantly to the availability of condensation sites; thus, we note that the anticipated impact of these defects on the results quoted elsewhere in this paper is minimal.

Adsorption experiment
The multilayer particle beds are then tested under a series of different relative humidities (RH).Relative humidity control was realized by bubbling dry air through a temperature-controlled water bath which is held at varying temperatures to achieve different saturation humidities.After bubbling through the water bath, the humidified air is fed directly to a glove box.For the water-uptake measurements, particle samples are first dried under a vacuum inside an opened airtight bag for 24 h.During this time, the humidity chamber is also in operation to ensure that it reaches the desired relative humidity prior to testing.After this period, the (still opened) airtight bags containing particle samples are transferred into the RH-controlled chamber and allowed to accumulate condensate.After waiting a sufficient time for the sample weight to stabilize (typically an additional 24 h), the bags are sealed, and measurements of mass gain are made using a Mettler Toledo MS105DU balance.

Results and discussion
In an effort to characterize the pore spaces between self-assembled particle layers, we have evaluated three different close-packed structures: Body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal closepacked (HCP).Based on the approximation that the particles have a relatively narrow size distribution, we assume them to be monodispersed within each cubic unit.Additionally, we assume that each single cubic unit cell within these packed structures can be used to inform the local values of the pore size distribution, as shown in figure 4. Ultimately, these assumptions allow us to calculate a representative inter-particle pore size distribution from the measured particle size distributions of the samples such that we may infer theoretical water uptake values.It is important to note that the pore size distributions do not have a sharp cut-off because we explicitly account for the polydispersity of each particle and utilize the aforementioned assumptions to translate this polydispersity into pore size variation across the whole sample.One can see that the pore size distribution shifts slightly with different packing structures, but that the particle size sets the order of magnitude of each slate of pore distributions.For simplicity of calculation, the critical pore sizes were estimated based on the maximum curvature radius between the adjoining particles in each unit-cell structure and approximating each interparticle pore as a perfect sphere with the requisite radius of curvature of the pore (See the complete calculation of the maximum curvature in the Supporting Information).While one notes that, in reality, the edges of the pore contain very sharp geometries, (eventually leading to a contact point between particles), the approach taken here allows simply theoretical predictions that represent a reasonable (but systemically larger value for) the critical humidity for the onset of capillary condensation within the particle systems.Moreover, the magnitude of this system approximation error increases as the particle size (and concomitantly the pore size) increases.
As discussed in the previous chapter, the theoretical predictions of capillary condensation in our closely packed structures are calculated based on the Kelvin equation (equation ( 1)) and informed by measurements (and modeling) of the structure of our samples.The Kelvin equation is derived from the Laplace equation, which expresses the pressure difference across the curved surfaces of a liquid.According to the thermodynamic relations, the Kelvin equation leads to a relationship between the relative pressure (relative humidity in the water vapor case) and the principal radius of the curved surface.Based on the Kelvin equation, the saturation vapor pressure will be depressed in the vicinity of the concave surfaces.Thus, for these geometries the relative humidity required for water vapor to condense is lower than that observed near a flat surface.That is, there will be a critical relative humidity value-which is lower than the 100% -that corresponds directly to the radius of the pores, and when the relative humidity is equal to or greater than this critical value the pore be filled with condensed water.Since, in reality, the inter-particle pore is not spherical, but instead has tight spacing at the edges of the pores, the requisite relative humidity for condensation will be lower than predicted in these areas (and the process of filling will initiate in these areas of smallest curvature); however, given that the bulk of the pore volume can be well-characterized by our approximations, the relative humidity that is associated with each specific pore size is indicative of the point at which the bulk of the water uptake is observed.
Using this approach and based on the calculated pore size distribution of each sample (obtained from combining the unit cell analysis with the particle size distribution), we are able to predict the capillary condensation process from 0% relative humidity to 100% relative humidity for each of our experimental samples.The black and gray lines in figure 5 show the theoretical water uptake predictions calculated based on the FCC, BCC packing, and HCP packing scenarios (the solid line represents the FCC structure, the dashed line represents the BCC structure, and the dotted line represents the HCP structure).It is noteworthy that, owing to the lower packing density of the BCC structure, water uptake at 100% relative humidity exceeds that of both FCC and HCP structures.This is attributable to the increased interstitial space available within the BCC Figure 5.The theoretical and experimental values of water uptake for samples with different-size particles.The black and gray lines show the theoretical water uptake predictions calculated based on the FCC, BCC packing, and HCP packing scenarios (the solid line represents the FCC structure, the dashed line represents the BCC structure, and the dotted line represents the HCP structure).Water uptake results for samples with different particle size distributions are shown as green triangles.An example of an ideal sample with 10 nm monodisperse particles in both BCC (thick dashed line) and FCC (thick solid line) packing is shown in (a).
structure for water absorption.For comparison, the ability of the particle samples to induce capillary condensation is measured under different relative humidity using the previously described humidity-controlled glove box.For each sample, the measurements are taken under different relative humidities ranging from 15% to 95%.For each measurement, the sample is allowed to reach equilibrium over a one-day exposure.Furthermore, the samples are thoroughly dried under vacuum between steps to ensure no residue affects the results.The masses of all tested samples are measured pre-and post-exposure (with samples sealed in an airtight bag for transport between the humidity chamber and scale).The absorptive performance is quantified based on the mass of water absorbed relative to the mass of the silica particles on the wafer.
We should note that a control wafer sample with no silica particles is also tested under both 50% and 90% RH.This control experiment resulted in no measurable water uptake; thus, we can interpret all water uptake observed in figure 5 to be the result of capillary condensation specifically within the pore spaces that are created by the presence of the closely packed silica particles.In figure 5 the water uptake results for samples with different particle size distributions are shown as the green triangles in each separate plot.It can be seen that, despite the difference in sizes, almost all samples ultimately achieved the same level of water retention, around 40% ∼ 50%, near 100% relative humidity (due to essentially the same void fraction being observed for each sample).Thus, when all the pores are filled, the total mass of the water relative to the mass of the particles is similar in each case.Due to the relatively narrow size distribution, the water uptake values increase dramatically over a relatively short range of relative humidity, especially for the samples comprised of very small particles.Again, here we emphasize that in the ideal situation (i.e., with a matrix formed by monodisperse particles), the water absorption will be a step change.In figure 5(a), we show an example of an ideal sample with 10 nm monodisperse particles in both BCC (thick dashed line) and FCC (thick solid line) packing.With an increase in the mean diameter of particles, the onset of water uptake requires a higher relative humidity.Also, the water uptake increases more slowly for larger particles due to the broader particle size distributions (and the more considerable discrepancy between the sharp point contacts and the as-defined, 'critical' pore sizes).
In figure 5, for samples with smaller particle sizes, the experimental observations coincide well with the theoretical predictions.Notice that for larger particles, noticeable differences are evident between the experimental water uptake curves relative to those based on the theoretical predictions.In order to rationalize this discrepancy, it is critical to acknowledge the limitations inherent in the particle self-assembly technique used to create the ordered porous structures.Specifically, the overall packing structure of our sample does not exactly conform to any of the three structures we utilized in our theoretical predictions of the observed water uptake.This arises because, in large particle systems, the formation of a closely-packed structure cannot be perfectly achieved through mild disturbances, such as solvent evaporation.This limitation is due to the influence of additional factors, including unevenly distributed disturbances, friction, gravity, and surface tension, among others [37,38].
Furthermore, manipulating capillary condensation by altering the particle size distribution introduces other challenges.Despite our approximation, the pores created within these closely packed structures are not spherical; instead, the pores can be tetrahedral, octahedral, or even more complicated, depending on the relative positioning of the adjoining particles.Different pore geometries result in less precise onset humidities during capillary condensation Specifically, the more sharply curved portions of the pores will start nucleating water below the theoretical 'critical' relative humidity, that might be naïvely assumed based on particle diameter, due to the existence of the local curvature differences.In other words, as expected, our simple model always overestimates the relative humidity at which the onset of capillary condensation occurs.
Moreover, for samples that include larger particles (100s of nm), the discrepancy in size between the sharp contact points between particles and the 'bulk' pore sizes estimated from the inter-particle distances can differ by up to an order of magnitude.As such, the water uptake before the critical relative humidity is considerably more apparent in these cases.In other words, in cases involving larger particles, capillary condensation invariably occurs far below the 'predicted' relative humidity, because the 'pores' estimated based on particle size alone are dramatically larger than the nucleation sites found in the vicinity of interparticle contact points.
In addition to the challenges mentioned, other factors such as surface chemistry and particle shape also impact the capillary condensation behavior of the fabricated porous materials.These factors can alter the wetting properties and the effective pore geometry, further complicating the predictive modeling of water uptake.
Despite these challenges, it is heartening that we predict, and observe, a plateauing of the water uptake at intermediate humidities for the samples containing small particles.However, we do not either predict or observe such a plateau for the larger-particle samples (other than the consistent evidence of a lower uptake value plateau that is observed near 50% relative humidity and which we attribute to condensation exclusively at particle contact spots).Finally, one can also notice that the ultimate water uptake near 100% relative humidity is always higher than the theoretical value.This discrepancy is particularly evident for the 200 nm sample, as may be anticipated from the image included in figure 3 (left).We anticipate that this is due to the filling of the aforementioned 'large pore' defects created from our evaporative self-assembly technique, which are ultimately filled at very high relative humidity and lead to more water uptake near 100% relative humidity.This discrepancy underscores the importance of considering large pore defects in the predictive modeling of capillary condensation in fabricated porous materials.

Conclusions
In summary, a closely packed particle matrix with controllable pore sizes is fabricated using a simple evaporative particle self-assembly method.This porous material with controllable sizes is tested for water uptake isotherms and compared against a simplified theoretical model.The experimental results show substantial agreement with our theoretical predictions of capillary condensation, particularly for systems comprised of very small particles.Given that our model is based solely on the Kelvin equation and the geometric characteristics of the structure, our work not only shows a facile way to control capillary condensation for future applications, but also lends further support to the applicability of the Kelvin equation for systems containing sub-10 nm geometries.
By demonstrating a user-friendly way to control capillary condensation for a wide range of applications, our research provides new insights and possibilities in the field of porous materials.Furthermore, the successful application of the Kelvin equation to these advanced materials lends additional support to its relevance and adaptability in designing tailored porous structures.This pioneering work paves the way for the development of next-generation materials with enhanced properties, offering a solid foundation for future research and practical implementations in various industries.

Figure 1 .
Figure 1.Particle size distributions of synthesized particles obtained using the Zetasizer, where the sizes are represented as the mean radius for each group to describe the central tendency of the size distributions: (a)10 nm, (b) 19 nm, (c) 82 nm and 86 nm, (d) 216 nm and 289 nm.

Figure 2 .
Figure 2. The atomic force microscopy image of a closely packed multilayer of 35 nm particles deposited on a glass slide.

Figure 3 .
Figure 3.The scanning electron microscopy images of the 200 nm particles at the center (a) and the edge (b).