Amphiphilic nanopores that condense undersaturated water vapor and exude water droplets

Macroscopic water droplet formation on amphiphilic nanoporous PINFs

Amphiphilic nanoporous PINFs are prepared by infiltrating hydrophobic polymers into the interstices of disordered packings of hydrophilic NPs. We select PE and SiO₂ NPs, both of which are commonly available and used, as the hydrophobic polymer and hydrophilic NPs, respectively. As depicted in Fig. 1A, a bilayer film composed of a PE layer atop a SiO₂ NP packing is heated above the melting temperature (T_m) of the PE, so that the fluidized PE undergoes capillary rise infiltration (CaRI) into the interstitial voids of the NP packing (15, 16, 26–29); when the volume of PE is less than the void volume of the NP packing, a nanoporous PE-infiltrated SiO₂ NP film is obtained as the PE distributes throughout the NP packing (26, 27). The integrity of the NP packing remains intact during PE infiltration due to strong interparticle interactions (28, 31), ensuring that both the microstructure and macroscopic topology remain nearly unchanged across different ϕ_PE, as shown in Fig. 1B and figs. S6 and S7. The ratio of the hydrophilic and hydrophobic regions within the nanopores can be simply tuned by changing the relative thicknesses of the NP and polymer films. ϕ_NP remains 0.64, whereas ϕ_PE can be varied from 0 to 0.36 by adjusting the volume of infiltrating PE. As ϕ_PE increases, the porosity of the films decreases (Fig. 1B and fig. S6). The value of ϕ_PE after CaRI can be determined on the basis of the effective refractive index (n) of the PINF composed of PE, SiO₂ NP, and void.

Remarkably, when these amphiphilic nanoporous PINFs are exposed to high yet subsaturating conditions [i.e., relative humidity (RH) < 100%], macroscopic water droplets appear spontaneously on the film surfaces without the need for cooling, as illustrated in Fig. 1C and shown in Fig. 1D. Specifically, macroscopic water droplets isothermally form when the NP size is ≤22 nm, RH is >~90%, and ϕ_PE ranges from 0.05 to 0.35. As will be described in detail later, a moderate value of ϕ_PE is necessary to create amphiphilic nanopores, consisting of hydrophilic SiO₂ NP surfaces and hydrophobic PE surfaces, which not only induce capillary condensation but also spontaneously exude the condensate. In addition, high RH and small NP size are required to achieve a sufficiently high extent of capillary condensation.

Characterization of amphiphilic nanoporous PINFs

Our prior work has shown that if the volume of polymer infiltrated is smaller than that of the void in the NP packing, then the polymer spreads throughout the thickness of the packing and forms capillary bridges between two adjacent NPs, analogous to what would be expected when capillary condensation takes place in NP packings (26, 27, 32). This topographical and chemical heterogeneity could have complex effects on the extent of capillary condensation and the wettability of the film surfaces, which may in turn affect the formation of macroscopic water droplets on the film surfaces. We assess the wettability of the film surface by placing a sessile droplet of water on 250-nm–thick films and measuring the contact angle. As ϕ_PE increases, the water contact angle increases from ~0° toward 100°, corresponding to those on SiO₂ NPs (33) and a pure PE film (fig. S8), respectively, as presented in Fig. 2A. This trend suggests that the uppermost surface of the films becomes enriched in PE as ϕ_PE increases due to the infiltration of PE into the interstitial pores. While the contact angle increases for all three NP sizes, the ϕ_PE at which the water contact angle sharply increases is smaller for larger NP sizes.

To better understand this sharp upturn in the contact angle with increasing ϕ_PE and its dependence on the particle size, we perform simulations using a coarse-grained lattice model, which is described in detail in both Materials and Methods and texts S1 to S3. In this model, NPs are packed in an ideal face-centered cubic lattice, with the (111) plane normal to the z axis, and the infiltration of PE is carried out using the lattice model. Using enhanced sampling simulations, we systematically increase the extent of polymer infiltration into the NP films and quantify it using the ratio, ϕ_PE/ϕ_void, of the amount of polymer in the system to the maximum amount of polymer that the PINF can hold. The evolution of the polymer morphology as a function of polymer infiltration is shown in Fig. 2B, and it highlights that the polymer occupies highly curved regions between NPs, forming capillary bridges that connect the NPs. As the polymer fraction is increased, the capillary bridges become larger and leave behind smaller pores. Moreover, we find that for smaller ϕ_PE/ϕ_void, the polymer preferentially occupies regions in the interior of the film, and polymer appears at the top surface of the film (Fig. 2C) only at much higher ϕ_PE/ϕ_void. We believe that this preference stems from the ability of polymer to bridge several NPs at once in the interior of the film. As shown in Fig. 2D, the polymer fraction at the surface remains negligible until 50 to 75% of the total void volume in the film is occupied by the polymer. By estimating the relative areas of polymer, NP, and void at the uppermost layer of the surface for a range of ϕ_PE/ϕ_void, and using the Cassie and Wenzel equations, we estimate the effective contact angle, θ_eff, shown in Fig. 2E; the top-down snapshots used for extracting the relative areas are shown in fig. S4A. The water droplet contact angle mirrors the sharp upturn in the surface polymer fraction with increasing ϕ_PE in agreement with the experiments (Fig. 2A). In addition, as ϕ_PE increases during polymer infiltration, capillary bridges are initially formed across neighboring NPs. As shown in Fig. 2F, upon increasing ϕ_PE further, the pores are not filled by the polymer uniformly (i.e., homogeneous filling), but rather in a sequential manner with some pores being filled completely and others remaining largely empty (i.e., heterogeneous filling).

To elucidate the impact of the NP diameter, D, on the observed wetting behavior (Fig. 2A), we simulate a set of films with the same thickness, H, but different number of particle layers, H/D. This approach seeks to capture the different number of particle layers that are present in 250-nm–thick films described in Fig. 2A. As shown in Fig. 2D, as the films contain fewer layers of NPs, the upturn in the surface polymer fraction is observed at lower ϕ_PE, which, in turn, results in the corresponding upturn in θ_eff occurring at lower ϕ_PE (Fig. 2E). This reduction in the threshold of ϕ_PE stems from the fact that for PINFs with fewer NP layers, the film interior comprises a smaller fraction of the overall film and is therefore filled by polymer at smaller values of ϕ_PE. The leftward shift of the inflection in θ_eff is consistent with experimental observations for the 77-nm NP films, which has a much smaller particle-layer thickness (H/D = 3.2) than the 7- or 22-nm NP films (H/D = 35.7 and 11.4, respectively). Experimental observations further support the dependence of water contact angle on the number of particle layers, showing that the water contact angle increases on 22-nm NP films as the film thickness decreases at a given ϕ_PE (fig. S9). Additional results showing the inputs to the Cassie equation and further information about the aspect ratio dependence are included in text S3.

Capillary condensation in amphiphilic nanoporous PINFs

We investigate capillary condensation in the amphiphilic nanoporous PINFs (7-nm NP, thickness ~ 250 nm) using environmentally controlled in situ ellipsometry with gradually increasing RH. We model the films using a one-layer Bruggeman effective medium approximation (EMA) accounting for the four components (water, void, PE, and SiO₂) that would be present, as depicted in fig. S10A, and monitor the changes with respect to RH. As RH increases, the effective refractive indices of the films increase [fig. S10B (a)] and become saturated at ~70% RH [fig. S10B (b)], regardless of ϕ_PE. The increase in the refractive index indicates an increase in the amount of water (n = 1.33) that fills the voids (n = 1.00) via capillary condensation, as schematically illustrated in Fig. 3A (a). The saturated refractive index values correspond to those estimated when the voids are completely filled by water, indicating that capillary condensate occupies the entire voids [Fig. 3A (b)]. Because of the higher refractive index of PE (n = 1.48) compared to those of water and void, the films with greater ϕ_PE have higher refractive indices when the pores are fully filled with the condensate (fig. S10B). It is interesting that capillary condensation takes place readily and that the condensate can completely saturate the pores at a similar RH despite varying amounts of hydrophobic PE present in the pores.

More intriguingly, at RH above 70%, the thicknesses of the films show apparent increases (pronounced at ≥90% RH), whereas that of the bare SiO₂ NP film remains almost unchanged, as presented in fig. S10C. The apparent increase in thickness suggests a mass uptake after the voids of the films have been completely saturated with water, possibly due to water build-up on the film surface, as schematically illustrated in Fig. 3A (c). The increase in thickness is notable for the films with ϕ_PE values of 0.10 and 0.13, possibly representing larger amounts of water that are present on the surface.

To accurately capture the differences in the refractive indices of the water-filled film and the water layer above the film, we reanalyze our ellipsometry data using a two-layer model, which comprises the four-component Bruggeman EMA, described above, as a bottom layer with a fixed thickness and a top Cauchy layer composed of water and air (Fig. 3B). The two-layer model enables us to independently monitor the changes in the film itself and the overlayer that forms (Fig. 3A). The actual compositions of the films at different values of RH can be estimated from the refractive index changes in the bottom layer, as displayed in Fig. 3C. ϕ_NP is kept constant at 0.64, while the sum of ϕ_NP, ϕ_PE, ϕ_water, and ϕ_void is kept at 1.00. When the ratio of ϕ_water to ϕ_void, indicating the fraction of void space occupied by water, is plotted as a function of RH, these curves overlap, as shown in Fig. 3D, implying that the increase in ϕ_PE does not considerably affect the extent of capillary condensation. To explain why pores with different polymer fractions condense water at similar RH, we must first understand the mechanism of polymer infiltration. As illustrated by the configurations shown in Fig. 2B and the analysis in Fig. 2F, as ϕ_PE increases during polymer infiltration, the pores are not filled uniformly (by the polymer), but rather in a sequential manner with some pores being filled completely and others remaining largely empty. Consequently, as ϕ_PE increases, the number of empty pores decreases, but the chemistry and surface topography of the pores remains the same. Because the critical RH at which condensation occurs is dictated by the chemical and topographical characteristics of the pores (and not how many pores are available), the critical RH tends to be independent of the polymer fraction, ϕ_PE.

To shed light on the mechanistic pathways that result in the uptake of water vapor into our amphiphilic nanoporous films, we use coarse-grained simulations that are described in detail in Materials and Methods. In particular, we use a lattice model of water that captures the interfacial tension of liquid water, and we parameterize the interactions of water with the polymer and the NP to recapitulate the experimentally determined water droplet contact angles on PE and SiO₂, respectively. Our coarse-grained model should thus capture the pertinent interfacial thermodynamics that drive capillary condensation. Additional simulation details are included in Materials and Methods and texts S1 and S2. By tracking the average amount of water as a function of RH, we obtain a wetting curve akin to the experimental system, as shown in Fig. 3E. Much like the experiments (Fig. 3D), we see a sigmoidal profile with a large shift near 70% RH. We can relate the regions on the curve with the different configurations seen along the wetting profile (Fig. 3F), with the initial capillary wetting occurring at low RH values, the inflection corresponding to filling the interstitial voids happening at intermediate RH values, and large values of the RH corresponding to the engulfment of the top layer of NPs. The considerably sharper sigmoidal curve predicted by the simulations as compared to that obtained experimentally (Fig. 3D) likely is due to the ordered array of NPs that is used in the simulations. The experimental system has a broad distribution of pore sizes and shapes that contribute to gradual filling of the pores as RH is increased (Fig. 3D).

Spontaneous droplet formation due to the expulsion of water from the amphiphilic nanopores

To confirm the presence of water on the top surfaces of these amphiphilic nanoporous PINFs at high RH, we observe the surfaces using optical microscopy. We find that water appears on the surface in the form of droplets rather than as a film, as shown in Fig. 4A. Representative videos showing the appearance and evolution of water droplets over time are provided in movies S1 to S3. We study the time-dependent evolution of the water droplets, as shown in Fig. 4A. Initial water droplets that are observable under optical microscopy (~1 μm in size) appear within a few seconds after being exposed to 97% RH. Micrometer-sized droplets increase in size over time, and some of them undergo coalescence as indicated by white arrows (Fig. 4A). We do not observe any second droplets forming and growing after the first few seconds. As a result, the number of droplets decreases, while their average and total volumes increase over time, as presented in Fig. 4 (B and C). The volume of a single droplet can be estimated using the contact angle and the radius of the droplet, assuming a spherical cap. We use the contact angle determined using sessile water drops as reported in Fig. 2A. This value is consistent with the contact angle of droplets that form spontaneously at ~100% RH (fig. S11). When water droplets reach a certain size, the system reaches a steady state. As the volume of voids decreases with increasing ϕ_PE, the growth and coalescence of water droplets are slowed down.

An investigation into the relationship between film thickness and the volume of water droplets provides key insights into the origin of these droplets on the surface. As depicted in Fig. 4 (D and E), we find that the total volume of the droplets increases linearly with the film thickness. This increase correlates with the volume of the nanopores inside the films, as the pore volume scales linearly with film thickness (i.e., a thicker film contains a proportionally greater volume of nanopores available for condensation). Such a trend suggests that the water droplets primarily emerge from the internal voids of the films, not directly from the surrounding vapor. Therefore, we believe that these droplets originate from water condensed in the PINF voids, not from direct condensation on the surface. The fact that the locations where droplets form are random (fig. S12) support this mechanism. Specifically, we believe that these micrometer-sized droplets result from the growth and coalescence of numerous nanodroplets seeping out of nanovoids at the top surface of the film. We also believe that any nanoscopic droplets that form on the surface can either evaporate or contribute to the growth of larger micrometer-sized droplets through the process of Ostwald ripening with interconnected nanopores (fig. S13) (34), facilitating the process. These factors, we believe, result in the inverse relationship between the number and average volume of droplets presented in Fig. 4B.

The relationship between the total volume of water droplets and ϕ_PE forms a bell-shaped curve (Fig. 4F), peaking at ϕ_PE = 0.13. The ϕ_PE at which the largest total volume of water droplets is obtained remains at 0.13, even with a different NP size (fig. S14). This maximum indicates that an optimal condition exists for the maximum volume of water that forms on the surface under this mechanism. The observed nonmonotonic trend in the total volume of water droplets spontaneously forming on the film surface implies the presence of two conflicting factors. We observe two opposing trends as ϕ_PE increases: an increase in the number of droplets but a decrease in the average volume of each droplet, as shown in Fig. 4 (G and H). We hypothesize that the reduction in the pore volume, limiting the space available for water to condense into, is counterbalanced by the increase in hydrophobicity of the pores, which promotes the formation of water droplets via exudation from the pores. Furthermore, no water droplets are observed below the RH of ~90% for the films with 7-nm SiO₂ NPs, as displayed in Fig. 4I. The fact that the apparent thickness measured using ellipsometry increases above 70% RH (fig. S10C) suggests that water droplets that are invisible under optical microscopy may exist in between 70 and 90% RH. The critical RH at which water droplets cease to be visible decreases with both NP size (Fig. 4I) and temperature (fig. S15), highlighting the crucial role of capillarity in the formation of water droplets. Further insight into the critical RH and critical NP size for which water droplets are no longer visible can be found in text S4.

Universality of the spontaneous water droplet formation

To interrogate whether the formation of water droplets on these amphiphilic nanoporous PINFs depends on the inherent properties of the polymer, we prepare PINFs with other common hydrophobic polymers, polystyrene (PS) and polydimethylsiloxane (PDMS). The water contact angles on pure PS and PDMS films are comparable to that on a pure PE film (~100°), as shown in Fig. 5A. The contact angle of PS on SiO₂ surfaces is ~20° (35), similar to that of PE, and accordingly, the water contact angles on the PS-infiltrated SiO₂ NP films as a function of ϕ_PS are also similar to those on the PE-infiltrated SiO₂ NP films (Fig. 5A). Likewise, we observe spontaneous formation of water droplets on nanoporous PS-infiltrated SiO₂ NP films with the overall trend mirroring that of the nanoporous PE-infiltrated SiO₂ NP films, as shown in Fig. 5 (B and C). In contrast, when PDMS is used, the number, average size, and total volume of droplets considerably decrease (Fig. 5, B and C). Compared to PE and PS, PDMS has a lower contact angle of 10° on SiO₂ surface due stronger interactions (fig. S16); for a given value of ϕ_polymer, the exposed SiO₂ surface (i.e., SiO₂ surface that is not covered by the polymer) is smaller when PDMS is infiltrated in place of PE or PS, leading to higher water contact angles on the PDMS-infiltrated SiO₂ NP films. The smaller unmodified SiO₂ area yields weaker driving force for capillary condensation, resulting in a smaller amount of water condensed in and exudated from the amphiphilic nanoporous PINF.

Materials

Monodisperse PE [number-average molecular weight (M_n) = 78,000 g mol⁻¹, P2861-E] and PS (M_n = 80,000 g mol⁻¹, P8500-S) were purchased from Polymer Source (Dorval, QC, Canada), and ultraviolet (UV)–curable PDMS (KER-4690 A and B) was purchased from Shin-Etsu Chemical (Tokyo, Japan). For the SiO₂ NP aqueous dispersions with different NP sizes, Ludox SM-30 (7 nm) and TM-50 (22 nm) were obtained from MilliporeSigma (St. Louis, MO, USA), and Snowtex ST-YL (60 nm) was generously provided by Nissan Chemical America (Houston, TX, USA). Isopropyl alcohol (Certified ACS Plus), NaOH solution (1 N/certified), decahydronaphthalene (99%), and toluene (high-performance liquid chromatography grade) were purchased from Thermo Fisher Scientific (Pittsburgh, PA, USA). Potassium sulfate (ACS reagent, ≥99.0%), sodium carbonate (ACS reagent, anhydrous, ≥99.5%), potassium chloride (ACS reagent, 99.0 to 100.5%), and rhodamine B (≥95%) were obtained from MilliporeSigma. Ethanol (200 proof) was purchased from Decon Laboratories (King of Prussia, PA, USA).

Preparation of the amphiphilic nanoporous PINFs

The stock SiO₂ NP aqueous dispersions were diluted with deionized (DI) water (18.2 megohm∙cm) to ~15 weight % (wt %), and then the diluted dispersions were sonicated for at least 2 hours to ensure a homogeneous dispersion of the SiO₂ NPs, followed by filtration using hydrophilic syringe filters with a size cutoff of 0.45 μm (09-720-005, Thermo Fisher Scientific) to remove potential NP agglomerates. PE was dissolved in decahydronaphthalene at 170°C at a concentration range of 0.2 to 2.0 wt %. PS and UV-curable PDMS (a 1:1 mixture of the precursors A and B) were dissolved in toluene at concentration ranges of 0.2 to 2.0 wt % and 0.4 to 4.0 wt %, respectively.

A Si wafer (452, UniversityWafer, South Boston, MA, USA) was cleaved into ~1.5 cm–by–1.5 cm pieces, and then the surfaces of the cleaved Si wafers were rinsed with isopropyl alcohol and DI water, followed by oxygen plasma (PDC-32G, Harrick Plasma, Ithaca, NY, USA) treatment for 5 min to eliminate any potential residual organic contaminants. SiO₂ NP packings with desired thicknesses were deposited onto the cleaved Si wafers by spin coating the prepared SiO₂ NP dispersions using a spin coater (WS-400BZ-6NPP/Lite, Laurell Technologies, North Wales, PA, USA). The spin coater was run for 1.5 min at 7000 and 3000 rpm to achieve thicknesses of ~130 and ~250 nm, respectively, and thicknesses beyond these values were achieved with multiple coatings in succession. Polymer layers with desired thicknesses were then coated on top of the SiO₂ NP films by spin coating the prepared polymer solutions at 5000 to 7000 rpm for 30 s. The PE-SiO₂ NP bilayer films and PS-SiO₂ NP bilayer films were annealed in a vacuum oven (Model 281A, Thermo Fisher Scientific) at 180°C for 12 hours to induce the infiltration of the polymers into the interstices of the SiO₂ NP layers via CaRI (15, 16, 26–29). The UV-curable PDMS layers spontaneously infiltrated into the interstices of the SiO₂ NP layers via CaRI during and after spin coating (16). The UV-curable PDMS-infiltrated SiO₂ NP films were exposed to UV light (illuminance of ~100 mW/cm², wavelength = 365 nm) for 30 s and then aged for 24 hours to allow for full curing of the PDMS.

Film characterization

The thickness and effective refractive index of the films were determined using a spectroscopic ellipsometer (SE-2000, Semilab, Budapest, Hungary). The ellipsometer measures parameters Ψ and ∆, which correspond, respectively, to the amplitude ratio and the phase difference of the complex reflection coefficients of light-polarized parallel and perpendicular to the plane of incidence (40). The measurements were carried out at 75° incidence angle in the spectral range of 1.3 to 4.4 eV (282 to 989 nm). The RH of the air surrounding the films was controlled using a humidity chamber (operated by a mass flow controller to ensure the ratio of dry and wet air inlet) compatible with the ellipsometer (fig. S18A) and widely used for ellipsometric porosimetry (41). All measurements were performed at 25°C. The measured optical data were analyzed by the Spectroscopic Ellipsometry Analyzer (SEA) software from Semilab to extract the thicknesses and effective refractive indices. The infiltrated NP layer was modeled using a four-component Bruggeman EMA composing of SiO₂ NP (ϕ_NP fixed at each step), PE (ϕ_PE fitted at RH = 0, and kept constant for subsequent steps), void (ϕ_void fitted at all steps), and water (ϕ_water fixed to 0 at RH = 0 and fitted at all other steps). For two-layer models, the top layer was assumed to consist of water and air with fitted thickness. To find the fit global optimum at each RH measurement point, Price’s algorithm (a controlled random search method built into the SEA software) was used with max iterations of 3000 to achieve small mean square errors (<5).

The microstructure of the films was visualized using a high-resolution SEM (JSM-7500F, JEOL, Tokyo, Japan). Before analysis, the films were coated with 4-nm Ir layers using a sputter coater (Q150TES, Quorum Technologies, Lewes, UK) to prevent possible charging. The SEM was operated under the conditions of 5-kV electron beam voltage and 20-μA emission current. To confirm the interconnectivity of the voids, the transport of a dye (rhodamine B), incorporating ethanol solution within the films, was monitored using an upright fluorescent microscope (Axioplan 2, Zeiss, Oberkochen, Germany) mounted with a charge-coupled device (CCD) camera (MU1400B, AmScope) and mercury lamp (HBO 100, Zeiss) in the reflection mode. Contact angles were determined using a goniometer (Attension, Biolin Scientific, Gothenburg, Sweden).

Water droplet formation on the amphiphilic nanoporous PINFs

To expose the amphiphilic nanoporous PINFs to desired RH, the films were placed in a custom polycarbonate chamber fabricated using a 3D printer (ProJet 6000 HD, 3D Systems, Rock Hill, SC, USA), and the chamber was filled with various saturated salt solutions and then sealed with a high vacuum grease (DC976, Dow Corning, Midland, MI, USA) (fig. S18B); the water vapor pressure in equilibrium over saturated salt solutions is lower than that over pure DI water and depends on the type of salts (42, 43). Potassium chloride, sodium carbonate, potassium sulfate aqueous solutions, and DI water were selected to achieve 86, 92, 97, and 100% RH, respectively. A RH of ~60% was naturally present on a cloudy summer day in Philadelphia (PA, USA). The top-view images of the condensed water droplets on the films were observed using an upright optical microscope (Euromex, Arnhem, Netherlands) mounted with a CCD camera (MU800, AmScope, Irvine, CA, USA) in the reflection mode. The side-view image was obtained using a custom optical microscope in the transmission mode. All measurements were performed at 20° ± 0.2°C maintained by an air circulation system unless otherwise noted. The temperature of the films was controlled using a heating/cooling unit (THMS350V, Linkam Scientific Instruments, Salfords, UK) when necessary.

The alternating amphiphilic nanoporous PINF-superhydrophobic surface mimicking the body of Namib beetle was fabricated by introducing amphiphilic nanoporous PINFs in the form of patches on a superhydrophobic surface. To prepare a superhydrophobic surface, a hydrophobic Teflon sheet was sanded using an electro coated abrasive paper (grit 400, Lanhu, Beijing, China); the given roughness led the surface to the Cassie-Baxter state (44). The stock SiO₂ NP aqueous dispersions (Ludox SM-30) was diluted with DI water and ethanol to 0.5 vol % (ratio of DI water to ethanol = 8:2), and then the diluted dispersion was treated in the same manner as described above. The prepared dispersion was sparsely deposited on the superhydrophobic surface by pipetting with a volume of 0.5 μl. After the deposited dispersions are completely dry, a PE layer with a desired thickness was coated on the superhydrophobic surface with SiO₂ NP patches by spin coating 4 wt% PE solution in 170°C decahydronaphthalene at 5000 rpm for 30 s. The resulting film was annealed in the same manner as described above.

Simulations

Both polymer infiltration into the NP films and the capillary condensation of water into the PINFs were simulated using a coarse-grained lattice gas model, which is based on the work of Vaikuntanathan et al. (45) and which captures the liquid-vapor surface tension and the power spectrum of capillary wave fluctuations. The model is described by the Hamiltonian

where the first two terms capture the properties of bulk fluid (either polymer or water), the third term is the accounts for interfaces (NP-polymer, NP-water, and polymer-water), and the last term represents a harmonic biasing potential, which allows us to drive the system to the desired degree of wetting. Every site i on the cubic lattice can be either liquid-like (n_i = 1) or vapor-like (n_i = 0). Thus, the first term, which sums over neighboring lattice sites, favors liquid-liquid contact, while the second term favors liquid-like sites akin to a chemical potential. Following Vaikuntanathan et al. (45), a lattice site length of λ = 0.184 nm and an interfacial energy term of ε = 1.35 k_BT are used. To obtain 100% RH or liquid-vapor coexistence, we choose μ = −3ε. To account for the presence of interfaces, a per-site energy contribution of the form n_iφ_i drives or inhibits the presence of liquid in each cell, where a positive value of φ_i inhibits the presence of liquid, while a negative value promotes i to be liquid-like. The φ_i field was constructed by integrating the energetic contribution of NPs and/or polymers to each cell using a force field based on the 9-3 Lennard-Jones potential, as described in detail in text S1. All surfaces are parameterized to properly capture the relevant contact angles between each phase (text S2). The biasing potential, U_κ,N*(N_v) = 0.5κ(N_v − N*)², is used to influence the total number of liquid-like lattice sites, N_v, inside an observation volume, v, of interest with κ being the spring constant and N* the set point of the biasing potential. This model additionally captures the qualitative features of condensation, with nucleation barriers being reduced with increasing surface hydrophilicity (46), as illustrated in fig. S19.

To explore NP films with different extents of polymer infiltration, a set of biased simulations with different values of N* were used, and the resulting structures of the polymer within the NP film were then analyzed to estimate the degree of filling, ϕ_PE/ϕ_void. The resulting structures of the infiltrated polymer were also used to estimate the polymer contribution to the φ_i field. Capillary condensation of water was then simulated (in the μVT ensemble at T = 300 K), similarly spanning the full range of wetting, and free energies were computed using sparse sampling (47, 48). To estimate the actual number of waters, the density of water was assumed to be ρ_L = 33 waters/nm³_. Simulation snapshots were generated using OVITO Basic 3.17.12 (49).

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