Supercritical Fluid Adsorption to Weakly Attractive Solids: Universal

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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Supercritical Fluid Adsorption to Weakly Attractive Solids: Universal Scaling Laws Miroslaw S. Gruszkiewicz,*,† Gernot Rother,*,† Lukas Vlcě k,† and Victoria H. DiStefano†,‡ †

Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6110, United States Bredesen Center, University of Tennessee, Knoxville, Tennessee 37996-3394, United States

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‡

ABSTRACT: Adsorption of light hydrocarbons (C1−C3) confined in the pores of silica aerogel (nominal density 0.2 g/ cm3) was studied experimentally and by computer simulation. Five isotherms of ethane (25.0, 32.0, 32.3, 33.0, and 35.0 °C) were obtained over a temperature range including the critical temperature of the bulk fluid (Tc = 32.18 °C) to ∼220 bar by the vibrating tube densimetry (VTD) method, which yields unique information impossible to obtain by excess sorption measurements. Two isotherms (35 and 50 °C) of methane adsorption in SiO2 aerogel were measured far above its critical temperature, Tc = −82.59 °C, at 0−150 bar using a gravimetric method. The experimental pore fluid properties of methane, ethane, propane, and CO2 (obtained earlier by VTD for the same SiO2 aerogel sample) show behavior conforming to the principle of corresponding states over the entire covered reduced temperature ranges from Tr = 0.976 to Tr = 1.696. The magnitudes of the excess adsorption maxima and the reduced fluid densities where the maxima occur for C1−C3 hydrocarbons and CO2 are similar functions of the reduced temperature. This behavior indicates that, at least under the conditions covered in this work, the differences in molecular size and shape have only minor impacts on the solid−fluid interactions underlying the adsorption behavior. These findings are supported by large-scale GCMC simulations of a lattice gas confined in the pores of a computer representation of the SiO2 aerogel framework, yielding excess adsorption isotherms in good agreement with experimental measurements, as well as the underlying microstructures of the adsorbed fluids.

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INTRODUCTION The interactions of liquids and dense supercritical fluids with solid surfaces and pores are different and less thoroughly examined than analogous interactions of gases and vapors, which have been the topic of most adsorption studies since the 18th century. In both cases the behavior of the fluid is the result of the balance between competing fluid−solid and fluid− fluid forces; however, gas or vapor adsorption usually results in a considerable reduction of fluid volume (increase of its density), similar to bulk fluid condensation, even at temperatures significantly higher than the temperature of condensation of bulk fluid at the same pressure or higher than the critical temperature Tc. In this case, the difference between the densities of bulk and adsorbed phases is so large that the effect of the changing density (and volume) of the adsorbed phase on the pore volume available to the free fluid can be neglected over a wide range of pressures. This is in contrast to dense fluid adsorption, where the bulk and adsorbed phases may have comparable densities, and therefore the pore volume fractions they occupy and derived quantities, such as the absolute amount of pore fluid or “gas-in-place”, are difficult to determine by traditional adsorption methods without knowledge of the adsorbed phase properties. Estimates of the volumes (densities) of the adsorbed phase, which in general © XXXX American Chemical Society

vary with temperature and pressure, are needed even to obtain accurate excess sorption by the traditional adsorption methods, particularly at supercritical conditions.1 For these reasons and because of the concomitant lower compressibility of dense fluids relative to gases and vapors, stronger solid−liquid interactions may be required to significantly modify the density of a liquid or supercritical fluid, and accordingly, the effects of confinement on fluid properties may be in general more subtle than in gases and vapors. An important exception is the broadly defined critical region, where a substantial maximum of excess adsorption vs pressure or bulk fluid density is commonly observed.2−7 These peaks do not occur at lower pressures in vapor or gas sorption isotherms, which are often characterized by excess adsorption continuously increasing with increasing pressure.8 In industrial and natural environments, many largescale processes involve dense fluids, and the implications of fluid−solid interactions (sorption) may be significant, since even small variations of the adsorbed phase density could entail significant changes of the total amount of fluid stored in a porous material. In industrial applications, these differences Received: May 8, 2018 Revised: June 17, 2018 Published: June 18, 2018 A

DOI: 10.1021/acs.jpcc.8b04390 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

dependent corrections are needed to obtain pore fluid density as a function of pressure. The absolute magnitude of the density is calibrated by the assumption of the convergence of pore fluid and bulk fluid densities at high pressure, and the final excess adsorption result is scaled by the estimated total volume of the pores, based on the difference between the effective and true (skeletal) volume of the solid.3 As a macroscopic method, the VTD approach can only provide the mean density of the pore fluid contained within the active (oscillating) part of the adsorbent sample placed inside the vibrating tube cantilever. In a simplified interpretation, this means that the measured density is an average of the densities of two fluid phases: the adsorbed phase found close to the pore walls, and the free or bulk fluid found farther from the pore walls, whose density in large pores may be practically unaffected by the force field of the solid. With decreasing average pore size, as the adsorbent becomes nanoporous or microporous, the fraction of this unadsorbed phase will decrease, and the measured average density will approach the properties of the adsorbed phase. Because the silica aerogel adsorbent investigated in this work has a relatively open pore system, with 7−9 nm voids between silica strands, with only moderately large specific surface area (∼300 m2/g), an accurate determination of the fraction of the adsorbed phase and its density could be derived from additional results for adsorbents similar to those reported here but with increasingly tighter structures (more microporous). At some small pore size depending on molecular sizes and fluid−fluid/fluid−solid interactions in each adsorbate−adsorbent pair, practically 100% of the pore fluid could be considered as the adsorbed (confined) phase. Thus, the “true” density of the confined fluid measured in the limit of very small pores could then be used to estimate the fractions of pore fluid in larger pores containing also significant fractions of free fluid on the basis of the experimental data for mean pore fluid density in such systems. The models developed for prediction of the effects of confinement and describing the properties of pore fluids have been largely focused on pore systems with well-defined geometries, such as cylindrical or slit pores, that is, unlike silica aerogels with ill-defined geometries. With growing interest in adsorption of liquids in complex heterogeneous natural adsorbents such as gas shales and other rocks containing significant fractions of clays and organic matter, the attention has been shifting to adsorbents with disordered (random) structures. The measured densities of CO2 in silica aerogel as a function of pressure previously reported by us3 were well represented by López-Aranguren et al.11 using the first-order equations12 based on a mean-field lattice gas approach. In this model, the adsorbent is represented by a random cubic matrix of sites occupied by the solid with a relatively low site density p (high porosity) and with two energy coupling parameters: one for the fluid−matrix and another for fluid−fluid interactions. One of the advantages of this model is the continuous transition from representation of the properties of a pure fluid (density of solid sites p = 0) to the fluid confined in random mesoporous structures (p > 0), with the accuracy comparable to that achieved by equations of state of the Peng−Robinson type. With this model it is then possible to predict the shift of the critical properties of the pore fluid relative to the bulk fluid as a function of the adsorbent porosity and the fluid−lattice coupling parameter. One of the reasons for the selection of experimental conditions close to the critical temperature of ethane was to enable validation of

may be relevant to the design of fluid mixture separation processes,9 while in the field of subsurface geochemistry, predictive knowledge of fluid adsorption is critical for modeling of the behavior of geologic reservoirs and operation of such energy technology processes as hydrocarbon extraction from shales, geothermal heat mining, or geological CO2 storage. The goal of this work is to improve our understanding of the interactions between light hydrocarbon fluids and porous materials by studying sorption as a function of the properties of various fluids, solid surfaces, and pore system geometries. Computer modeling facilitates development of generalized predictive capabilities of sorption processes over wide ranges of temperature, pressure, and materials. New experimental results of methane and ethane adsorption in synthetic silica aerogel are presented and analyzed in conjunction with previously reported propane and carbon dioxide sorption data.3 Two different experimental approaches were applied: vibrating tube densimetry (VTD) for ethane and gravimetric excess sorption using an electromagnetic suspension balance for methane adsorption measurements. We have shown3 that for the adsorbent used in this work, these methods give consistent results, with possible differences at the conditions that pose the biggest challenges to experimental accuracy: in the critical region of maximum compressibility and at the highest densities. Since the VTD and gravimetric approaches are based on different principles, their agreement confirms the methodology adopted for calibration of VTD measurements of density, the derivation of excess adsorption of dense fluids at elevated pressures from VTD and gravimetric experimental data, and the validity of the corrections and assumptions applied in both methods. The VTD approach measures the difference between the eigenfrequencies of oscillation of the U-tube cantilever, under vacuum and when the fluid is filling the pores, which is proportional to the inert mass of the fluid and therefore represents the changes of the density of the pore-confined fluid. This manner of direct determination of the mean pore fluid density is the feature that makes VTD distinct and complementary to the gravimetric and volumetric methods, thus providing an independent cross-validation of experimental excess sorption data. In the gravimetric approach, the excess adsorption is determined as the difference between the measured weight of the porous adsorbent sample in bulk fluid and its buoyancy. In gases or vapors the buoyancy effect is a small and slowly variable correction to the weight, but experimental excess adsorption for dense fluids may need to be determined as a difference between two larger quantities. Since it is difficult to assess the buoyancy correction accurately,10 particularly so for significantly microporous adsorbents, such as clays, the uncertainty of gravimetrically measured excess adsorption tends to increase with increasing fluid density. Likewise, volumetric methods are also affected by the limited accuracy of the void space volume determination, as the volume occupied by the adsorbed phase of unknown density changes with pressure. These known weaknesses of the established adsorption methods in high-pressure applications are the reason for interest in the VTD approach. VTD measurements utilize a void-free sample of the adsorbent, tightly filling the volume of the sample tube, without macroscopic (much larger than average pores) void spaces filled with bulk fluid. As a consequence, no first-order (essential with respect to the experimental error) pressureB



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such predictions made with this “equation of state for confined fluids” and other models. In this work, instead of a simple random cubic lattice, we apply a model based on a computer reconstruction of the fractal silica aerogel matrix13 using cluster−cluster aggregation algorithms,14,15in a procedure analogous to that used in our previous study.6 Compared to the random matrix approach, it allows us to investigate the structural details of the fluid density distribution and, potentially, compare the modeling predictions with experimental data from small angle neutron or X-ray scattering techniques (SANS or SAXS). The fluids are represented as lattice gas models, which are chosen for their simplicity and the fact that they belong to the same universality class as the real fluids.16

SIMULATION METHODOLOGY Aerogel Matrix Reconstruction. To reconstruct the aerogel matrix, we followed the procedure established in our previous study of CO2 adsorption in silica aerogel, but with aerogel of higher density (0.2 g/cm3 vs 0.1 g/cm3).6 Here, we only summarize the main steps and simulation parameters. To create a realistic reconstruction of the aerogel matrix, we used a two-step procedure based on the chemically limited cluster− cluster aggregation (CLCA) algorithm19 followed by surface annealing. In our CLCA simulations, the basic units (spherical silica particles) diffuse through space and may form a bond with the sticking probability p = 0.9 when they collide, a value that was found to produce best agreement with small angle neutron scattering (SANS) spectra.6 The simulations started from random configurations of nonoverlapping spheres, whose sizes were drawn from the Gaussian distribution with mean, μ = 4 nm, and standard deviation, σ = 1 nm, while the pore volume fraction was kept at the experimental value of 0.9. To keep the simulations consistent with our previous results, the size of the simulation box was set to 0.53 μm3. To anneal the silica surface and create a framework for the subsequent lattice gas simulations, we divided the simulation box into a regular grid with a cubic unit cell whose volume was set to values corresponding to the “excluded volume parameter” b of the van der Waals equation of state for each fluid. For methane b = 0.04278 L/mol, ethane b = 0.0638 L/ mol, propane b = 0.08445 L/mol, and carbon dioxide b = 0.04267 L/mol. Each lattice unit was then assigned to either silica or pore space, depending on its overlap with the silica beads. The annealed effect observed in the TEM image was achieved by reducing the surface energy through moving under-bonded surface sites (with one or two nearest neighbors) to more favorable positions. Grand Canonical Lattice Gas (LG) Simulations of Fluid Sorption. The distribution of fluids within the reconstructed aerogel matrix was modeled using grand canonical Monte Carlo (GCMC) simulations with an LG model implemented on a simple cubic lattice with periodic boundary conditions. The number of grid points N for a particular system depends on the size (volume) of each cubic cell derived from the van der Waals b parameter. For methane N = 10243, ethane N = 8963, propane N = 8003, and CO2 N = 10243 lattice sites. The Hamiltonian, H, of the system can be written as

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EXPERIMENTAL SECTION Methods. The experimental details of the vibrating tube apparatus modified for measurements of pore fluid densities have been described earlier.3 The isotherms of adsorption/ desorption of ethane were measured by gradually increasing the pressure of ethane in the vibrating tube in small steps by means of a manual positive displacement pump from vacuum (99.999%), certified to contain volumetric fractions of less than 0.9 ppm of other hydrocarbons and less than 0.1 ppm of each of the following: N2, O2, H2, CO + CO2, and moisture. The methane was Airgas research grade (purity >99.99%).

H = −μ∑ nif − wff ∑ nif njf − wfs∑ nif nks i

i,j

i,k

(1)

where μ is the fluid chemical potential, wff is the energy of fluid−fluid interaction, wfs is the energy of fluid−solid interaction, nfi is the occupancy (either 0 or 1) of the fluid site, nsk is the occupancy of the solid site, the first summation runs over all lattice sites, and the remaining two summations run over all nearest neighbor pairs of lattice sites. The correspondence between the LG model and real fluids was established by using reduced units for temperature, pressure, and density, that is, by effectively matching their critical parameters. The reduced fluid−fluid interaction parameter was set to unity, wff = 1, while the fluid−solid parameter was optimized to approximate the experimental adsorption isotherms for temperatures close to critical. We found that setting wfs = 1.2 for all considered alkanes resulted in a good match with experimental excess sorption isotherms. C


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Figure 1. Isotherms of density of pore ethane measured using the vibrating tube method and those of bulk ethane calculated from the equation of state.18 The density difference (along the vertical axis) between these isotherms, caused by a pressure shift (along the horizontal axis), represents the excess adsorption, shown below in Figure 2.

bulk ethane calculated from its equation of state18 and both the adsorption and desorption isotherms of the pore fluid. At 25 and 32 °C two phases should coexist in bulk ethane at the corresponding liquid−vapor equilibrium pressures. Indeed, these two isotherms feature sections of vertical straight lines and the resulting two discontinuities of the slope, corresponding to the two phases in equilibrium. These discontinuities are sharper (better defined) at the high density point, representing the liquid phase, than at the low density point, representing the vapor phase. The existence of the corresponding discontinuities of the slope in the pore fluid isotherms in Figure 1 at 25 and 32 °C indicates that the pore fluid is also subcritical at these two temperatures and may even be still subcritical at 32.3 °C. This means that no decrease of the critical temperature due to fluid confinement was found in this work, or if present, the shift was too small (a small fraction of 1 K) to be detected by our method relying on detection of discontinuities in the slopes of the pore fluid isotherms. This conclusion is consistent with our earlier results3 for propane in the same pore system, which indicated no more than 1.7 K Tc depression at about 97 °C. The small magnitudes of the critical temperature shifts are a consequence of relatively large pore size of the open pores in the silica aerogel, in comparison with the results for prevalently microporous adsorbents reported in the literature. Much greater effects have been observed in other pore systems, e.g., over 40 K Tc shift for CO2 in microporous Vycor glass.20

For comparison, in the study of CO2 adsorption we found a good match for wfs = 0.9.6 The higher value of wfs for alkanes than for CO2 does not necessarily imply their stronger interactions with silica surface but rather the higher relative strength of the fluid−solid interactions with respect to the respective fluid−fluid interactions. In each cycle of the GCMC simulation, the occupancy nfi of each site i was changed with the “switching” probability Pswitch i determined from the following expression: Piswitch

ÄÅ É l ÅÅ o o ij yzÑÑÑÑ| o o Å o j z Å o ÅÅ− β(1 − 2n f )jjμ + w ∑ n f + w ∑ n szzÑÑÑo = mino 1, exp m } j z Å Ñ i ff j fs k o o j z Å Ñ o o j z Å Ñ o o ÅÅÅ Ñ o j / i k k {ÑÑÖo Ç n ~ (2)

The fluid chemical potential was varied between values corresponding to bulk densities 0.1ρc and 0.9ρc. At each chemical potential the system was first equilibrated until the total fluid density between MC cycles did not change by more than 10−6ρc and then was simulated for additional 100 equilibrated cycles to calculate average densities at each lattice site. These averages were collected and analyzed.

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RESULTS AND DISCUSSION Figure 1 shows the density of pore ethane in silica aerogel as a function of pressure at five temperatures (25.0, 32.0, 32.3, 33.0, and 35.0 °C), of which the first two are subcritical for the bulk fluid (Tc = 32.18 °C). The diagrams include the isotherms for D


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noise (Figure 2B), there is a good agreement between the experimental and simulation results. The most important feature of Figure 2 is the apparently uniform pattern of variability of the excess reduced density isotherms with reduced temperature for different fluids. Since the features of the isotherms in terms of the reduced properties are similar, different fluids behave similarly as adsorbents in the sense of the principle of corresponding states. This is illustrated in Figures 3 and 4, which show the coordinates of

The excess adsorption measured by experimental macroscopic methods is defined in terms of the excess density of the pore fluid, which is caused by fluid adsorption to the pore walls. This excess density is the difference between the confined and bulk phase fluid densities shown in Figure 1. It is clear from Figure 1 that the maximum of excess density (the maximum of the difference between the isotherms on the vertical axis in Figure 1, also shown explicitly in Figure 2

Figure 2. Excess reduced density of the confined fluid as a function of the reduced density of bulk fluid: (A) experimental data; (B) calculated from the lattice gas model with optimized fluid−solid interaction parameters, wfs.

Figure 3. Maximum of the excess reduced density of the confined fluid as a function of the departure of the reduced temperature from critical: (red box) propane; (green dot) ethane; (purple up-triangle) methane; (blue ×) carbon dioxide. The inset shows the temperature range close to the critical temperature.

below) is primarily the consequence of a simple pressure shift of the pore-confined fluid isotherm toward lower pressures, relative to the bulk fluid isotherms. The pressure of the vapor− liquid phase transition in subcritical pore fluids, or, at supercritical conditions, the pressure at the corresponding high compressibility regions, visible in Figure 1 as the steepest slope sections of the curves, is lower than the corresponding pressure in bulk fluid. This pressure shift alone, even without a significant concomitant densification effect, is responsible for the well-known maximum of the excess adsorption associated with the maximum compressibility of the fluid, which is located in the critical region. The magnitudes and shapes of these maxima are additionally affected by the different magnitudes of the densification effect for different adsorbate−adsorbent pairs at the corresponding sections of the isotherms. Note that the density of the pore fluid follows the behavior of the density of the bulk phase at the same T and corresponding P conditions, and accordingly it changes with T and P. It is not in general close to the density of the liquid at normal boiling point or to the critical density, as has been often assumed in the analysis of excess sorption results. Therefore, using any of these constant densities to calculate absolute adsorption from excess sorption data may lead to significant error.1 The isotherms of excess reduced density for ethane and methane measured in this work, together with the isotherms for propane and carbon dioxide obtained earlier, are plotted in Figure 2A as a function of the reduced densities of the bulk fluids. The reduced densities of both confined and bulk fluids were calculated as ρr = ρ/ρc where ρc is the critical density of bulk fluid. The critical densities used in the calculation were 0.16266, 0.207, 0.220, and 0.4676 g/cm3 for CH4, C2H6, C3H8, and CO2, respectively. Figure 2B shows the matching results of the computer simulations of sorption as described above at selected temperatures. While the heights of the peaks near the phase boundary and the critical temperature are difficult to determine precisely in simulations because of the high level of

Figure 4. Reduced density of the bulk fluid at the maximum of the excess reduced density of the confined fluid as a function of the departure of the reduced temperature from critical: (red box) propane; (green dot) ethane; (purple up-triangle) methane; (blue ×) carbon dioxide. The inset shows the temperature range close to the critical temperature.

the maxima visible in Figure 2A as a function of Tr − 1, the departure of the reduced temperature from its critical value (unity), where the reduced temperature Tr = T/Tc, with Tc = 190.564, 305.33, 369.825, and 304.128 K for CH4, C2H6, C3H8, and CO2, respectively. The maximum of excess density, corresponding to the maximum of excess adsorption, increases as the temperature decreases from supercritical values (low density supercritical fluid or gas) to Tc and below (high density supercritical fluid or vapor), as shown in Figure 3. The region of the steepest change is clearly visible in the inset in Figure 3, focusing on the narrow temperature range close to Tc. To the extent that the data for all four fluids confined in the silica aerogel appear to fall onto a single curve, these pore fluids are behaving in a similar manner in the sense of the corresponding E


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marker of strongly changing fluid properties”23,24 and an “extension of the first-order coexistence line in the supercritical region.”21 At Pr < 3 and Tr < 1.2 the Widom line is quite well represented by the critical isochore.22 At Pr > 3 or Tr > 1.2 the “pseudoboiling” phenomenon in supercritical fluid becomes less pronounced, and accordingly the excess adsorption maxima become almost negligible relative to those in the vicinity of the critical point. Both the experimental and the simulation results show that the maxima of excess density or excess adsorption of the pore fluid occur along the phase coexistence line and its extension into the supercritical fluid, approximately along the critical isochore, and they are highest in the critical density region. It should be noted that a number of other criteria of changing fluid properties and transitions in the supercritical region have led to definitions of additional border lines, which can be determined by neutron scattering experiments and by MD simulations. Changes of fluid density and structure, interpreted as droplet formation in supercritical CO2, were found recently in the vicinity of the Widom line by SANS methods.25 The well-known existence of the maximum of the effective isothermal compressibility of any fluid occurring at the vapor− liquid coexistence boundary (the boiling line) or its extension (the Widom or pseudoboiling line) at a certain pressure (density), in combination with the pressure shift of these lines caused by fluid−solid interactions, is thus sufficient to produce maxima in the isotherms of excess adsorption vs pressure of any adsorbent−adsorbate pair. The present results show that the pore fluid properties of the C1−C3 alkanes and CO2 confined in synthetic silica aerogel are similar in the sense of the principle of corresponding states. This observation is consistent with the conclusion of Banuti et al.24 that the position of the Widom line in various fluids (including, among others, hydrocarbons to C6 and CO2) adheres well to the extended three-parameter corresponding states principle (including the Pitzer acentric factor ω). Thus, for molecules with relatively small acentric factors, the simple two-parameter principle of corresponding states will be also obeyed to a good approximation. Besides the relatively small values of the acentric factors for methane, ethane, propane, and CO2, another reason for this result is the lack of strong specific interactions between silica aerogel and these fluids. Measurements at lower subcritical temperatures might show greater deviations from similarity, as vapor−liquid coexistence curves and the properties of liquids in general increasingly deviate from the principle of corresponding states with decreasing reduced temperature. In addition to the modification of the corresponding P, T, ρ conditions in pore fluids, which could potentially follow scaling laws adequate to the complexity of the fluid−fluid and fluid− solid interactions, other effects can additionally affect the behavior of excess adsorption isotherms. For example, changes of the properties of some adsorbents can be induced by the pore fluid, such as adsorbent swelling, which may be anisotropic, area common phenomenon in microporous solids such as clays, coals, or zeolites. Such phenomena would potentially cause significant deviations from the classical scaling laws, as specific size, shape, and reactivity considerations become important for fluid−solid chemical interactions and pore sizes comparable with molecular diameters. Lattice Gas Simulations. All fluid simulations were performed in the same silica aerogel matrix, whose representative detail is shown in Figure 5. The simple lattice

states principle. The same observation can be made on the basis of the plots of the bulk fluid density at the maximum of excess confined fluid density, shown in Figure 4, analogous to Figure 3. The maxima of excess adsorption occur at subcritical densities of the bulk fluid, which reach their own maximum at the critical isotherm, as shown in Figure 4. The noteworthy apparent degree of adherence of these four fluids to the principle of corresponding states may originate from relatively weak, nonspecific fluid−solid interactions between nonpolar fluids and silica surface. This picture could change for polar and additionally hydrogen-bonded fluids interacting stronger with silica, such as water, and for carbonbased adsorbents interacting stronger with hydrocarbons. More pronounced departures from similarity would be also expected for significantly microporous sorbents. Some difference is noticeable in Figure 2 between ethane and carbon dioxide excess density isotherms at similar reduced temperatures (both experimental and simulation results). While the magnitudes and positions of the maxima of the two isotherms are almost identical, the ethane peak is wider, indicating a more pronounced densification of ethane extending beyond the critical region. The smaller effect of the solid wall on the density of CO2 may be the result of the stronger competing fluid−fluid interactions among strongly quadrupolar CO2 molecules, which is a hypothesis that can be tested by the simulations. The results of this work indicate that at elevated pressures and fluid densities, the properties of the pore fluid appear to follow quite closely the properties of the bulk fluid at the same conditions, modified by the fluid−solid interaction. The first order universal effects of this modification are (i) a shift of the vapor−liquid transition to lower pressure (the confined vapor condenses at a lower pressure than the bulk vapor), which is visible as a pressure shift toward smaller values of the pore fluid density isotherms relative to bulk density isotherms (Figure 1), and (ii) increased density of the fluid. The magnitude of this pressure shift is generally less pronounced for dense fluids than for gases as it should depend on the relative strength of fluid− fluid vs fluid−solid interactions, and therefore it should increase with increasing compressibility of the fluid. The density difference between adsorbed and bulk fluid (excess adsorption) depends on the slopes of the fluid density isotherms shown in Figure 1, which in turn reflect the isothermal compressibility κT of the fluid. The vertical sections of the isotherms in the subcritical region, corresponding to the first-order liquid−vapor phase transition, will clearly produce the greatest excess adsorption maxima as a result of even a very small pressure shift between the isotherms. Although the firstorder phase transition and coexisting phase equilibrium disappears above the critical point, there is still a sharp transition between vapor-like and liquid-like states in the supercritical region. In contrast to the subcritical vapor−liquid transition, this state change occurs over finite but increasingly wider ranges of temperature and/or pressure, as the pressure and temperature increase along the extension of the vapor− liquid coexistence line beyond the critical point, called the Widom line or the “pseudoboiling” line.21−23 The Widom line, a concept dating back about 60 years but “not widely appreciated”,21 has been defined as the locus of the maxima of the correlation length, but it is also a good approximation of the loci of the maxima of many macroscopic thermodynamic functions of a supercritical fluid, including the isothermal compressibility. Phenomenologically, it can be considered “a F


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investigated fluids, we present its predictions after optimizing the fluid−solid interaction parameters wfs in Figure 2B. In agreement with our previous study of CO2 adsorption in aerogel, the overall agreement between the experimental and simulated curves can be characterized as semiquantitative. The excess sorption peak positions are in good agreement with the experimental data, while their heights are underestimated by 0−15%. Testing a range of fluid−solid interaction parameters during optimization showed that the excess sorption peak heights are largely insensitive to the strength of fluid−solid interactions and reflect thus the fluid−fluid interactions and the shift in the critical point caused by confinement. In agreement with experiment, we can observe that the peaks of CO2 at 32 °C and ethane at 35 °C have the same height, but the former is narrower. Since the main difference between the two systems is in the weaker solid−fluid interactions relative to fluid−fluid interactions for CO2 (wfs = 0.9 vs wfs = 1.2 for ethane), the modeling results support the hypothesis ascribing the experimental narrower peak width to relatively stronger fluid−fluid interactions. Given the simplicity of the model and the large sensitivity of the peak height to the temperature variation near the critical point, the present level of disagreement is not surprising. Unlike the case of CO2 adsorption and in agreement with experiments, none of the simulated excess sorption curves exhibit negative excess sorption at high densities. In Figure 6, we show a representative density distribution of propane within the aerogel matrix at 97 °C and chemical potentials corresponding to three bulk densities: low, critical, and high. The two-dimensional sections through the porous matrix show that at low bulk densities the fluid adsorbs on the surface in a monolayer (case A), as could be expected. As the density approaches the bulk critical value, the fluid condenses in increasingly larger confined spaces, with longer-range density fluctuations (case B). At high chemical potentials (pressures), the fluid density in the middle of the pores reaches

Figure 5. Detail of the 0.2 g/cm3 silica aerogel structure as reconstructed using the CLCA algorithm.

gas model allows us to analyze the effect of the two competing forces of fluid−solid and fluid−fluid interactions and predict fluid distribution within the silica aerogel pores. While the former interactions can be thought of as localized at the fluid− solid interface and dominating fluid adsorption at conditions far from the critical point, the latter effect depends on the relative magnitudes of the confining pore size and the fluid correlation length and becomes increasingly more important for fluid sorption near the critical conditions, which are of main interest in the present study. To assess the ability of the model to reproduce experimental adsorption isotherms for the

Figure 6. 2D density profiles at 97 °C of lattice gas propane confined in silica aerogel matrix corresponding to different stages of pore filling. The letters (A, B, and C) indicate corresponding states in the 2D profiles and in the plot of excess sorption as a function of bulk density (bottom left). G


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the values typical for the adsorbed phase, and the distinction between the two phases disappears (case C).

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CONCLUSIONS The experimental and computer simulation results presented in this paper are consistent with each other, and they reveal the origin of the excess adsorption behavior observed previously in various cases of dense fluid adsorption. The most important conclusion is that the main features of excess adsorption isotherms, observed universally for a wide variety of adsorbent−adsorbate pairs at elevated pressure, can be explained by a simple pressure shift of the properties of the confined fluid in relation to the bulk fluid, combined with the necessary presence of the liquid−vapor phase change or its vestige (extension) in the supercritical region in the form of a continuous transition between liquid-like and vapor-like states. The resultant features of excess adsorption isotherms that can be predicted include the presence of the maxima of excess adsorption at elevated pressures, their magnitudes, and locations as functions of the properties of the fluid and P, T conditions. It was confirmed that the behavior of different simple fluids confined in the pores is similar in the sense of the principle of corresponding states. Deviations from the simple underlying relationships can be expected for more complex fluids and for very small pores, and therefore the methods described here should be applied next to a wider variety of pore systems and to fluid mixtures. Lattice gas simulations, which investigate the microscopic driving forces behind the observed adsorption isotherms, support the experimental results. The agreement of the model with experiment and the insensitivity of the excess adsorption to the value of the solid−fluid interaction parameter demonstrate that the observed behavior is a function of confinement rather than the strength of interfacial interactions. As illustrated in Figure 6B, the range of the density fluctuations in the supercritical fluid is comparable to the typical pore sizes (distances between silica threads) in aerogel. The solid matrix therefore limits the length scale of these fluctuations.

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AUTHOR INFORMATION

ORCID

Miroslaw S. Gruszkiewicz: 0000-0002-6551-6724 Gernot Rother: 0000-0003-4921-6294 Lukas Vlček: 0000-0003-4782-7702 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division.

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REFERENCES

(1) Sircar, S. Comments on the Size of the Adsorbed Phase in Porous Adsorbents. Adsorption 2017, 23, 917−922. (2) Donohue, M. D.; Aranovich, G. L. A New Classification of Isotherms for Gibbs Adsorption of Gases on Solids. Fluid Phase Equilib. 1999, 158, 557−563. (3) Gruszkiewicz, M. S.; Rother, G.; Wesolowski, D. J.; Cole, D. R.; Wallacher, D. Direct Measurements of Pore Fluid Density by Vibrating Tube Densimetry. Langmuir 2012, 28, 5070−5078. (4) Rother, G.; Krukowski, E. G.; Wallacher, D.; Grimm, N.; Bodnar, R. J.; Cole, D. R. Pore Size Effects on the Sorption of Supercritical H


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The Journal of Physical Chemistry C (24) Banuti, D. T.; Raju, M.; Ihme, M. Similarity Law for Widom Lines and Coexistence Lines. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2017, 95, 052120. (25) Pipich, V.; Schwahn, D. Densification of Supercritical Carbon Dioxide Accompanied by Droplet Formation When Passing Widom Line. Phys. Rev. Lett. 2018, 120, 145701.

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Supercritical Fluid Adsorption to Weakly Attractive Solids: Universal

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