Undulation Theory and Analysis of Capillary Condensation in

Article pubs.acs.org/JPCC

Undulation Theory and Analysis of Capillary Condensation in Cylindrical and Spherical Pores Poomiwat Phadungbut,†,‡ D. D. Do,*,† and D. Nicholson† †

School of Chemical Engineering, University of Queensland, St. Lucia, Brisbane, QLD 4072, Australia School of Chemical Engineering, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand

Downloaded by UNIV OF NEBRASKA-LINCOLN on August 27, 2015 | http://pubs.acs.org Publication Date (Web): August 20, 2015 | doi: 10.1021/acs.jpcc.5b04789

‡

ABSTRACT: The undulation theory, recently developed to explain the mechanism of the onset of condensation in an open end slit pore, is extended to investigate the effects of pore diameter, pore wall curvature, temperature, and surface strength on the condensation of argon in cylindrical and spherical pores. Using the concept of an undulating interface separating the adsorbed phase and the gas-like core, we can determine the mean thickness of the adsorbate phase and thus the mean radius of the gas-like core just before condensation. The radius of curvature of the core is used in the Cohan−Kelvin equation, modified to account for the contribution from the solid−fluid interaction (better known as the Derjaguin−Broekhoff−de Boer equation), to derive the interfacial energy parameter. For cylinders and spheres, this parameter is always greater than the bulk phase value, which is used in the unmodified Cohan−Kelvin equation but converges to the bulk value as the pore diameter is increased. For spheres, the energy parameter is smaller than in cylinders of the same diameter but is still higher than the bulk value for pore diameters less than 6 nm. This difference is attributed to packing effects in the 3D-confined spherical geometry. This shows the interesting interplay between packing effects in 3D-spherical confinement and the cohesiveness of the adsorbate enhanced by the solid−fluid interaction.

1. INTRODUCTION The capillary condensation phenomenon in the adsorption of gases in mesoporous materials at temperatures below the critical hysteresis temperature is characterized by a very sharp (firstorder) change in density in the adsorption isotherm and is accompanied by a hysteresis loop. This subject has been extensively investigated for several decades because of its importance in the characterization of porosity and pore size distribution1−3 in many porous solids with industrial applications. Advances in nanotechnology have led to the fabrication of highly ordered mesoporous materials such as MCM-41,4 SBA15,5 and porous alumina.6 The progress in molecular simulation has greatly improved our understanding of the fundamental physics of confined fluids, from which the correlation between the condensation mechanism and pore morphology, temperature, adsorbate, and surface heterogeneity can be established.7 In 1973, Everett and Haynes8 pointed out that there are two radii of curvature describing any curved surface and three possible configurations for meniscus curvature. They proposed a mesoscopic thermodynamic mechanism for capillary condensa© XXXX American Chemical Society

tion in cylinders, in which nucleation of the dense adsorbate phase originates from a wave-like undulation, which grows and transforms into a liquid bridge bounded by two hemispherical menisci. This led them to argue that the wave-like structure is the origin of the condensation process. This mechanism has been confirmed in a number of simulations.9−13 The original thermodynamic formulation for condensation in and evaporation from cylinders due to Cohan invoked only two forms of meniscus curvature: cylindrical and spherical with condensation occurring at the former and evaporation from the latter. The Cohan−Kelvin equations14 have been widely applied to describe condensation and evaporation in cylindrical pores, forming the basis for the determination of pore size distribution in the mesopore range. Cohan suggested that the difference in shape between the cylindrical meniscus during adsorption and the hemispherical meniscus during desorption was the reason for the Received: May 19, 2015 Revised: August 9, 2015

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DOI: 10.1021/acs.jpcc.5b04789 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

boundaries in the axial direction. The solid−fluid interaction energies for a cylindrical pore and a spherical pore were calculated as described in refs 31 and 32. The molecular parameters of a carbon atom in a graphene layer were σss = 0.34 nm and εss/kB = 28 K, and the surface density of the carbon atoms in the graphene layer was ρs = 38.2 nm−2. The cross-collision diameter and the reduced well depth of the solid−fluid interaction energy were calculated by the Lorentz−Berthelot mixing rule. Both pore models were composed of a single graphitic layer. 2.2. Grand Canonical Monte Carlo (GCMC) Simulation. In the GCMC simulations, we used 70 000 cycles in the equilibration stage and the same number for the sampling stage. Each cycle consisted of 1000 attempted displacements, insertions, or deletions with equal probability. In the equilibration stage, the maximum displacement length was initially set as half of the largest dimension of the simulation box and was adjusted at the end of every cycle to give an acceptance ratio of 20% for the displacement. For a given pressure, the chemical potential was calculated from the equation of state of Johnson et al.33 and was used as the input in the GCMC simulation. 2.3. Simulation Analysis. In order to obtain the local properties of the adsorbate, the pore space was divided into 1D bins along the radial direction. The 1D density distribution in cylindrical and spherical pores is defined by

existence of hysteresis. The classical (unmodified) Cohan− Kelvin equations are valid for large pores, where the adsorbate structure is largely that of the bulk liquid at the same temperature. However, in pores of nanodimensions these equations fail: (1) because the structuring effect, due to the external adsorbent field, extends throughout most of the adsorbate and (2) because the constraints of confinement (repulsive forces), especially in cylindrical and spherical geometries, impose limitations on molecular packing. In attempts to refine the classical equations, it was suggested, for example, in papers by Melrose,15 Foster,16 Barrett−Joyner−Halenda (BJH),17 and Kruk−Jaroniec−Sayari (KJS)18 that condensation occurred at the surface of an adsorbed layer whose statistical thickness, t, could be obtained from adsorption measurements on a nonporous solid with the same surface chemistry. These modifications extend the range of validity of the Cohan−Kelvin equation to smaller pores; however when pores are of nanodimensions, the contribution from the solid−fluid potential cannot be ignored. To this end, the work by Derjaguin19 and Broekhoff and de Boer (BdB) accounting for the solid−fluid potential at the interface separating the adsorbed phase and the gas-like core20,21 provides a significant advance in condensation theory. The BdB method has been tested successfully against a number of experimental data.22−27 Recently, Ustinov et al.28 further refined the BdB model for cylindrical pores by correcting the potential of a nonporous solid for the curvature of the pore wall, and Kornevet al.29 developed a mathematical theory in which the ideas of Derjaguin and the undulating structure of the adsorbate were both incorporated. In 2014, Fan et al.30 proposed a new undulation theory for the microscopic origin of condensation in slit pores. In this theory, when the distance between a maximum peak of the undulating interface at one wall and the corresponding peak on the opposite wall is twice the collision diameter of the adsorbate, condensation occurs because the two unduloid interfaces coalesce to form a biconcave liquid bridge, which then nucleates pore-filling. In this paper, using grand ensemble simulations, we extend the undulation theory to describe condensation in cylindrical and spherical pores and to study the effects of pore diameter, pore wall curvature, temperature, and surface strength on condensation. Using our simulation data, we determine the boundaries of the adsorbed film, of the fluctuation region, and of the gas-like core just before condensation and obtain values for the radius of curvature of the interface separating the adsorbed phase and the gas-like core. This radius of curvature is then used in the Derjaguin−Broekhoff−de Boer equation to determine the interfacial energy parameter.

ρ (r ) =

PNF(r ) =

liquid molar volume (L/kmol)

interface product, σνL (J·m/kmol)

87 100 110

14.69 11.28 9.00

28.78 30.62 32.36

4.227 × 10−4 3.455 × 10−4 2.911 × 10−4

⟨ΔN (r )2 ⟩ − ⟨ΔN (r )⟩⟨ΔN (r )⟩ ⟨ΔN (r )⟩

(2)

In locations where the adsorbate is very dense, close to the pore wall, the PNF is less than unity, but in the region where mass exchange is taking place between the adsorbed layer and the gaslike core, the PNF is greater than unity. In the ideal-gas limit, the PNF approaches unity. 2.4. Analysis of the Capillary Condensation. To analyze capillary condensation with the modified Cohan−Kelvin equation or the Derjaguin−Broekhoff−de Boer (DBdB) equation, the radius of curvature of the core needs to be determined, and it is necessary to specify a way to define this parameter in an unambiguous manner. Knowing the profile of the PNF versus the radial distance we can identify the thickness of the adsorbed layer, the thickness of the fluctuation region, and the size of the gas-like core. In a recent communication35 we showed that the boundary separating the adsorbate from an adjacent gas-like phase could be located by an equal area construction applied to a fraction of success profile measured during a simulation. The radius of curvature for a liquid meniscus (r*) can then be defined as the distance from the center of the pore to the midpoint of the fluctuation region. The radius of curvature, defined in this way, was used in either the modified Cohan−Kelvin equation or the DBdB equation. The Cohan− Kelvin equation can be written as

Table 1. Simulated Liquid−Vapor Surface Tension and Liquid Molar Volume36 of Argon Used in This Work surface tension (mN/m)

(1)

where ⟨ΔN(r)⟩ is the ensemble average of the number of particles in a bin bounded by [r,r + Δr], whose volume is ΔV(r). One measure that has been found useful in probing the nonuniformity of an adsorbate is the particle number fluctuations (PNFs). Along the radial direction this is defined as34

2. THEORY 2.1. Fluid−Fluid and Solid−Fluid Potentials. The argon adsorbate was modeled by the Lennard-Jones equation with the following parameters: σff = 0.3405 nm and εff/kB = 119.8 K. The adsorbents were cylindrical or spherical pores with graphitic pore walls. The cylindrical pores were infinite in length with periodic

temperature (K)

⟨ΔN (r )⟩ ΔV (r )

B


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Figure 1. Argon adsorption isotherms in (a) cylindrical and (b) spherical pores having sizes of 5 and 6 nm in diameter at 87 K (top) and their local density distribution and particle number fluctuation. I and II are indicated as dense adsorbate and fluctuation regions, respectively.

⎛P ⎞ R gT ln⎜ cond ⎟ = −σνLC ⎝ P0 ⎠

the absolute temperature; and C is the mean curvature of the meniscus separating the condensate and the gas-like phase. When the meniscus is cylindrical, C = 1/r*, and the above equation is referred to as the Cohan equation, for a spherical meniscus where C = 2/r* it is the Kelvin equation. Here r* is defined as (R − t) where R is the cylinder radius and t is an

(3)

where Pcond is the condensation pressure; P0 is the saturation vapor pressure of the bulk fluid; σ is the liquid−vapor surface tension; νL is the liquid molar volume; Rg is the gas constant; T is C


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Figure 2. Snapshots of argon molecules in cylindrical and spherical pores of 5 and 6 nm in diameter at 87 K at the pressure just before condensation.

adsorbate “statistical” film thickness. In eq 3 the surface tension and the liquid molar volume appear together, but it should be stressed that neither parameter necessarily corresponds to its value in a bulk fluid. Hereafter, we shall therefore refer to this product as the interfacial energy parameter, α. The Derjaguin and Broekhoff−de Boer (DBdB) models19,28 include an additional term to account for the solid−fluid potential and may be written in a general form ⎛P ⎞ R gT ln⎜ cond ⎟ = −σνLC + FSF(r *) ⎝ P0 ⎠

pores of macroscopic dimensions, the same behavior is observed in the nanodimensional pores considered here. Figure 2 shows a snapshot of adsorbed molecules at a pressure just before condensation in a 5 nm diameter cylinder and 6 nm diameter sphere and demonstrates the existence of an undulating interface in cylinders and spheres as was shown earlier for slit pores.30 Along the adsorption path, as previously found for the openended slit pore, the size of the gas-like core or core size diameter (Dcore) in cylindrical and spherical pores can be determined from the thickness of the adsorbed layer (dA) and that of the fluctuation region (dF) as

(4)

Dcore = D − (dA + dF)

where FSF(r*) is the contribution from the solid−fluid potential exerted on molecules at the boundary separating the adsorbed phase and the gas-like phase by the external field from the solid adsorbent. When this contribution is negligible, as in very large pores, the DBdB equation reduces to the modified Cohan− Kelvin equation. In Table 1 we list the interfacial parameters of the bulk liquid at various temperatures.

(5)

This can be used to define the condensation criterion at the point just prior to condensation as [Dcore = D − (dA + dF)]Pcond = Dcore|Pcond

(6)

where Pcond is the condensation pressure. 3.2. Effects of Pore Size. Figure 3 shows the effects of pore size on the argon adsorption isotherms at 87 K in cylindrical and spherical pores and the distributions of local PNF just before condensation. From Figure 3, we can draw a number of interesting observations: (1) When the pore size is increased, condensation occurs at higher pressure because a higher pressure is required before the thickness of the adsorbed layer is sufficient to leave a gas-like core small enough to induce condensation. It is also noted that the core size just before condensation is larger for a larger pore size, as expected because the condensation pressure is higher for a larger pore. This conforms to the DBdB theory that the radius of curvature of the interface just before condensation increases at higher condensation pressure. (2) The fluctuation region and the amplitude of the fluctuation just before condensation are larger as the pore size is increased. This is due to the strong curvature of the interface and the influence of the solid−fluid potential in smaller pores. Thus, in large pores molecules enter or leave the adsorbed phase more frequently. (3) For a given pore size, the core size in a sphere just before condensation is greater than in a cylinder, which is in turn greater than in a slit pore because condensation occurs at lower pressure at a high curvature interface. This will be detailed later. We see that for a given pore size the thickness of the adsorbed layer and that of the fluctuation region are deciding factors in the condensation and that the radius of curvature of the interface is related to this thickness and to that of the fluctuation region. Figure 4 shows the variation of thickness of these zones at the point just before the condensation as a function of pore size. The thickness of the adsorbed layer increases with pore size until the core reaches the critical size for condensation. What is quite intriguing is that the thickness of the fluctuation region also

3. RESULTS AND DISCUSSION 3.1. Undulation Theory for Condensation in Cylinders and Spheres. The undulation theory of Fan et al.30 was applied to describe the growth of the adsorbed layers in cylindrical and spherical pores as pressure is increased. The adsorption isotherms for argon adsorption at 87 K in a cylinder of 5 nm diameter and a sphere of 6 nm diameter are shown in Figure 1. Also shown in the same figure are the local density distributions and the local particle number fluctuations versus the radial distance at various points marked on the isotherms. Both isotherms exhibit a hysteresis loop of Type H1, in the IUPAC classification. A first-order condensation transition follows the growth of a metastable adsorbed layer, and the firstorder evaporation transition suggests that the desorption mechanism is one of cavitation. From the local PNF distribution plot, we identified three distinct regions: the dense adsorbed layer where the PNF is less than unity (I), the fluctuation region where the PNF is greater than unity (II), and the gas-like core where the PNF is close to unity (III). On the adsorption branch of the isotherm, adsorption proceeds by molecular layering on the pore walls as pressure is increased from an empty pore (points A, B, and C in Figure 1), and the volume of the adsorbed phase increases up to the condensation pressure when the pore is filled with condensate. The detailed molecular behavior of the adsorbed phase is revealed by analyzing the local PNF distribution versus distance shown in Figure 1. From this plot, at temperatures below the critical hysteresis temperature, the boundary of the adsorbed layer increases with pressure, and the fluctuation zone has an undulating interface, as first suggested by Everett and Haynes8 in their classical thermodynamic analysis of adsorption−desorption hysteresis in cylindrical pores. Although their theory applied to D


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Figure 3. Argon adsorption isotherms at 87 K (top) and local particle number fluctuations just before condensation (bottom) in (a) cylindrical and (b) spherical pores having various pore diameters.

As seen in Figure 4, the size of the gas-like core and the radius of curvature just before condensation increase with pore size, in contrast to the constant core size observed in slit pores.37 Our finding in this paper is consistent with the modified Cohan− Kelvin (mCK) equation, which accounts for the thickness of the adsorbed layer, since for larger pores a higher condensation pressure is required to build an adsorbed phase thick enough for

increases with pore size. This can be attributed to the effects of both the curvature of the interface and the influence of the solid− fluid potential. In small pores the solid−fluid (SF) potential well is deeper due to overlap of potential fields from the surrounding atoms, which tends to dampen the fluctuations, and also the curvature is greater. In the larger pores the interfacial curvature is less, and potential overlap effects are diminished. E


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Figure 4. Plots of dA, dF, radius of gas-like core, and radius of curvature for argon adsorption at 87 K in (a) cylinders having pore diameters of 3−9 nm and (b) spheres having pore diameters of 4−12 nm at the pressure just before condensation.

thickness of the adsorbed layer and the reduced condensation pressure versus pore size in Figure 5. For a given pore size, we see that the thickness of the adsorbed layer in spheres is lower than that in cylinders, while the condensation occurs at lower pressure. This suggests that condensation in spheres required fewer adsorbed layers than in cylinders because the SF potential is greater in spheres and therefore induces condensation sooner. Figure 6 shows the distance from the pore wall, of the boundaries of the adsorbed layer and of the fluctuation region, as functions of pressure within the hysteresis loops, up to the point just before condensation for a cylinder of 7 nm diameter and a sphere of 8 nm diameter. As pressure is increased, these boundaries move away from the pore wall. An interesting difference from an open-ended slit pore is that the thickness of the fluctuation region is increasing, and this can be attributed to the stabilizing effect of the stronger SF potential. This explains why the boundary of the fluctuation region in a sphere moves away from the wall faster as pore size increases than it does in a cylinder. 3.2.1. Analysis of the Cohan Equation for Condensation in Cylindrical Pores. Having determined the radius of curvature of the interface from the center of the pore to the midpoint of the fluctuation zone and the pressure at condensation from the

Figure 5. Thickness of adsorbed layer and relative pressure for argon adsorption at 87 K in cylinders and spheres at the pressure just before condensation.

condensation to occur and mCK states that the radius of curvature of the interface is greater at higher pressure. To further understand the nature of the adsorbed layer just prior to condensation in cylinders and spheres, we plotted the

Figure 6. Evolution of the boundary of the adsorbed layer and of the fluctuation zone, as a function of pressure for argon adsorption at 87 K. (a) In a cylinder of 7 nm diameter and (b) in a sphere of 8 nm diameter. Zones I−IV are identified as repulsion (I), dense adsorbate (II), fluctuation (III), and gas-like core (IV) regions. F


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Figure 7. (a) Comparison of simulation results determined by undulation theory and the Cohan equation for argon adsorption at 87 K in cylinders having pore sizes of 3−9 nm at the pressure just before condensation. (b) Interface parameter obtained from simulation results compared to the bulk value at 87 K.

Figure 9. Comparison of local density distribution of argon between a spherical pore of 5 nm, a cylindrical pore of 5 nm, and on a flat surface at 87 K at the points just before condensation.

Figure 8. (a) Comparison of simulation results determined by undulation theory and Kelvin equation for argon adsorption at 87 K in spheres having pore sizes of 4−12 nm in diameter at the pressure just before condensation. (b) Interface parameter obtained from simulation results compared to the bulk value at 87 K.

Figure 10. Gas−liquid interface parameter for argon adsorption at 87 K in cylindrical and spherical pores and for bulk phase argon, as a function of solid−fluid interaction at the radius of curvature for a liquid-like meniscus.

simulation, we can now test the applicability of the DBdB equation. In Figure 7a, we plot RgT ln(P0/Pcond) versus the inverse of the radius of the curvature; the dashed line is the classical Cohan equation with the surface tension and molar volume of the bulk liquid. Applying the DBdB equation, we derived the interfacial energy parameter, α, and plotted it as a function of pore diameter in Figure 7b. For the range of pore sizes considered, the interfacial energy parameter calculated from

the DBdB equation is greater than its value for the bulk liquid. This is in agreement with earlier reports in the literature.38 3.2.2. Analysis of the Kelvin Equation for Condensation in Spherical Pores. In Figure 8a we plot RgT ln(P0/Pcond) versus the inverse of the radius of curvature for adsorption in spheres. The interfacial energy parameter, calculated from the DBdB equation G


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Figure 11. Argon adsorption isotherms (top) and local particle number fluctuation (bottom) at a pressure just before condensation in (a) a cylinder of 5 nm diameter and (b) a sphere of 6 nm diameter from 77 to 110 K. The saturation pressure is calculated from the correlation given in ref 41.

cylindrical pore. We see that the second and higher peaks of the distribution for planar graphite and the cylindrical pore are closer to the surface than those for the spherical pore. These trends are all consistent with inefficient packing of molecules in the confined space of the sphere. 3.2.3. Effect of Solid−Fluid Potential on Derived Interfacial Parameters at 87 K. The interfacial energy parameter, characterizing the cohesiveness of molecules at the interface, is affected strongly by the solid−fluid potential. In Figure 10, we plot α as a function of the SF potential energy at the interface (r*) for cylinders and spheres. The graphs show that in addition to the role played by packing effects in small spheres the SF potential directly affects the cohesiveness of molecules at the interface separating the adsorbed phase and the gas phase. 3.3. Effects of Temperature. Figure 11 shows the temperature dependence of the isotherms and their PNF at the

and presented in Figure 8b as a function of pore size, now shows the opposite trend to cylinders and decreases with pore diameter for pores less than 6 nm but is, nevertheless, higher than the bulk liquid value. This can be attributed to the inefficient packing of molecules in a spherical confinement. As the pore diameter increases, efficient packing becomes more facile, and the cohesiveness of the interface due to the effect of the solid− fluid interaction becomes more dominant, resulting in an interfacial energy parameter that is greater than the bulk value. In even larger pores, the packing effect and the effect of the external adsorbent field both diminish, and the energy tends to the bulk value. To gain better insight into the packing effect in small pores, we present in Figure 9 the local density distribution for a 5 nm diameter spherical pore just before condensation and compare this with the equivalent distribution for a graphitic surface and a H


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Figure 12. Plots of dA, dF, the size of gas-like core, and radius of curvature in a cylinder of 5 nm diameter and in a sphere of 6 nm diameter as a function of temperature.

Figure 13. RgT ln(P0/Pcond) for argon adsorption at 87−110 K versus the inverse meniscus radius just before condensation pressure. Simulation results: symbols. Dashed lines: Cohan (cylinders) or Kelvin (spheres) equation using bulk liquid values for the energy parameter. (a) Cylinders having pore diameters of 5−12 nm and (b) spheres having pore diameters of 6−12 nm. The color schemes are black 87 K, red 100 K, and blue 110 K.

occurs; the core is filled with condensate; and the radius of curvature at the interface is nearly the same as the pore radius. In finite length pores, open to the external vapor phase, the evaporation proceeds by recession of the menisci whose radii of curvature are also nearly the same as the pore radius. This implies that, at the critical hysteresis temperature, the adsorption− desorption isotherm is reversible and there is a sharp jump in density associated with the condensation and evaporation in a core region whose size is close to that of the pore radius. This hypothesis will be studied in greater detail in the very near future. We have determined earlier, by extrapolation, that at 87 K the threshold pore diameter for capillary condensation in cylinders is about 10 nm, below which the DBdB theory must be used to account for the fact that the interfacial energy parameter is greater than that of the bulk. This means that the Cohan equation, even when modified for adsorbate thickness, is only applicable for pores having diameters greater than 10 nm. A question that now arises is what is the threshold pore size when the temperature is increased? Figure 13 shows the effects of temperature on the plot of RgT ln(P0/Pcond) versus the inverse radius of curvature at the relative pressure just prior to condensation for various pore diameters, as extracted from Appendix 1. We find that the threshold pore size increases with temperature for both cylinders (Figure 13a) and spheres (Figure 13b). 3.4. Effects of Strength of the Adsorbent Field. The effects of different adsorbent field strength on argon adsorption

point just prior to condensation for argon in cylinders and spheres with pore diameters of 5 and 6 nm. As expected, the pore density and the width of the hysteresis loop decrease due to the increase in thermal fluctuations as temperature is increased. In contrast to an open-ended slit pore,30 the condensation shifts to higher relative pressures, as also reported experimentally by Morishige et al.39,40 for adsorption in cylindrical pores. This occurs because the curvature of the pore wall dampens the undulation and stabilizes the adsorbed layer. To support this assertion, we plotted the thickness of the adsorbed layer and that of fluctuation just before condensation as a function of temperature as shown in Figure 12. For both cylindrical and spherical pores, as found in open-end slit pores earlier,30 the width of the fluctuation zone increases, and that of the adsorbed layer and of the gas-like core decreases, with increasing temperature due to the increase in thermal fluctuations at higher temperatures. We also found that the radius of curvature of the interface increases with temperature as predicted by DBdB theory. Further consideration of Figure 12 in the fluctuation region reveals that molecules are moving in and out of the adsorbed layer and that this can be regarded as a mass transfer zone whose width is zero at T = 0 K and which increases in thickness as temperature increases. This is why, when the temperature has reached the critical hysteresis temperature, there is a single adsorbed monolayer, while the fluctuation region spans most of the pore radius minus a monolayer. At this point condensation I


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Figure 14. Argon adsorption isotherms (top) and PNF just before condensation (bottom) in (a) a cylinder of 5 nm and (b) a sphere of 6 nm at 87 K with different surface strengths, one and three graphitic layers with εss/kB = 28 K and 10 K. I and II are indicated as dense adsorbate and fluctuation regions, respectively.

(1) The width of the fluctuation region increases with increasing pore diameter. In particular, the gas-like core diameter does not remain constant in contrast to our earlier study on slit pores. (2) For a given pore diameter, the size of the fluctuation zone in a cylinder is smaller than in a sphere because of the difference in interfacial curvature. (3) The width of the fluctuation zone increases with increasing temperature or adsorbent field strength, and that of the dense adsorbed phase and the gas-like region decreases. The interfacial energy parameter, α, which appears as σνL in the Cohan or Kelvin equations has been evaluated by plotting RgT ln(P0/Pcond) versus the inverse radius of curvature at the relative pressure just prior to condensation. For the range of cylinder and sphere diameters considered, the energy parameter is typically greater than that calculated using bulk phase quantities. In large pores, the fluid properties within the pore converge toward those of the bulk fluid, and the Cohan−Kelvin theories are excellent approximations. An interesting feature, not previously recognized, is that tight packing in smaller spherical pores results in disorder in the adsorbate and in a decrease in the interface parameter.

isotherm at 87 K in a cylinder of 5 nm diameter and a sphere of 6 nm diameter are shown in Figure 14. The figure also shows the PNF versus radial position at a pressure just before the condensation. For weaker adsorbents, condensation is shifted to higher pressure due to the slower growth of the adsorbed layer. It is notable, from the PNF plots, that the size of the fluctuation region and gas-like core both increase as the adsorbent becomes weaker. This is in qualitative agreement with the DBdB theory since a larger core radius is required before condensation can occur when the adsorbent strength is weaker.

4. CONCLUSIONS The undulation theory for capillary condensation under subcritical conditions, previously developed for slit pores, has been extended to cylindrical and spherical pores. A microscopic analysis of particle number fluctuation (PNF) has been used to interpret the mechanisms of condensation in these two pore geometries. The effects of pore diameter, temperature, and solid affinity on the wave-like fluctuation at the adsorbate gas-like interface are considered at a pressure just prior to condensation. The conclusions from this study may be summarized as follows: J


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Figure 15. Argon adsorption isotherms at 100 K (top) and their profiles of local particle number fluctuation just before condensation (bottom) in (a) cylindrical and (b) spherical pores having various pore widths.

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APPENDIX 1: EFFECTS OF PORE DIAMETER AT 100 AND 110 K

Argon adsorption isotherms and their profiles of local particle number fluctuation just before condensation are shown in Figures 15 and 16.

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AUTHOR INFORMATION

Corresponding Author

*E-mail address: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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ACKNOWLEDGMENTS

This research was made possible by the Australian Research Council and Suranaree University of Technology, whose support is gratefully acknowledged. The authors also thank the Thailand Research Fund for an award to PP through the Royal Golden Jubilee Ph.D. program; grant no. 1.C.TS/53/A.1. K


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Undulation Theory and Analysis of Capillary Condensation in

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