Mesoporous materials play an important role both in engineering applications and in fundamental research of confined fluids. Adsorption goes hand in hand with the deformation of the absorbent, which has positive and negative sides. It can cause sample aging or can be used in sensing technology. Here, we report the theoretical study of adsorption-induced deformation of the model mesoporous material with ordered corrugated cylindrical pores. Using the classical density functional theory in the local density approximation, we compared the solvation pressure in corrugated and cylindrical pores for nitrogen at sub- and super-critical temperatures. Our results demonstrate qualitative differences between solvation pressures in the two geometries at sub-critical temperatures. The deviations are attributed to the formation of liquid bridges in corrugated pores. However, at super-critical temperatures, there is no abrupt bridge formation and corrugation does not qualitatively change solvation pressure isotherms. We believe that these results could help in the analysis of an adsorption-induced deformation of the materials with distorted pores.

Mesoporous materials have a structure with pores in the range of 2 nm–50 nm and a large surface area and pore volume. All these properties make them attractive for numerous applications, among which are drug delivery,1 catalysis,2,3 and separation and selective adsorption.4 There are mesoporous materials with a disordered pore structure and broad pore size distribution (PSD) (porous glasses and gels) and ordered mesoporous materials with almost unimodal pore size distribution and well-defined pore geometry (MCM-41,5 SBA-15,6and porous aluminum7). The latter presents a nice, experimentally achievable, system for studying fluid properties in confinement, in particular, capillary condensation and related effects. The particular interest was attached to ink-bottle geometries convenient for studying pore-blocking and cavitation effects.8–10 The advances in the preparation of the materials with miscellaneous porous morphology facilitate these studies.7,11–14 The authors have shown that the adsorption/desorption mechanism is sensitive to the shape of the pore and wall roughness.

Both ordered and disordered materials deform during the fluid adsorption; this phenomenon is called adsorption-induced deformation, and it was intensively studied in the past decades both theoretically and experimentally.15 The solvation pressure in the pores, which is the driving force for the adsorption-induced deformation, depends on the pore size and shape, and many of the theoretical works focused on the description of solvation pressure in the pores with the regular pore shapes, slit,16–18 cylindrical pores,19–22 spherical,23,24 and hexagonally ordered system of rods.25 Furthermore, some works model adsorption-induced deformation of complex porous structures without any assumptions on the pore geometry. For instance, in Ref. 26, the lattice density functional theory was applied for the deformation modeling of sandstone. In another work,27 the authors developed a complex computational scheme for the calculations of the capillary stress in granular materials.

While the effects of pore surface heterogeneities, such as roughness or undulation, on adsorption isotherms were explored theoretically,28–32 the effect of pore surface heterogeneities on the solvation pressure (and deformation) has not been studied yet. This is the main focus of the current paper: here, we theoretically study the adsorption-induced deformation of the mesoporous material with corrugated cylindrical pores. The theoretical description is based on the model proposed by Gommes32 for the study of capillary condensation in such a system. Using the Derjaguin–Broekhoff–de Boer (DBdB)-like model, he demonstrated that the explicit account for the wall corrugations could explain the discrepancy between the pore width determined by x-ray diffraction analysis and adsorption measurements.32 We modified this approach for calculation of solvation pressure. Within our approach, we compare the solvation pressure in the cylindrical pore with those arising in the corrugated pore. Our results suggest that the formation of the liquid bridges could qualitatively change the strain isotherm.

The elastic properties of the material are treated as for a macroscopic object (which parameter however depends on the porosity and pore spatial arrangement), and the adsorption is described in the individual pore assumption. We also consider a sample that is disordered in the sense that there is no predominant orientation of the pores. Thus, we neglect the shape changes on the macroscopic scale so that the macroscopic deformation is isotropic. On the microscopic scale, each pore is represented as a corrugated cylindrical pore (see the schematic pore representation in Fig. 1).33 The pore profile is defined by the following equation:32 

Rp(z)=Rm+2σcos2πzlc,
(1)

where lc is the length of the corrugation, Rm is the average pore radius, and σ is the parameter defining the amplitude of the corrugation. The whole pore has length L, and we assume that LRm and σRm. As was shown in Ref. 32, a liquid bridge can occur on the adsorption or desorption branches of the adsorption isotherm.

FIG. 1.

The unique part of the corrugated pore, lc, is the corrugation length, Rm is the average radius of the pore, and 2σ is the amplitude of corrugation.

FIG. 1.

The unique part of the corrugated pore, lc, is the corrugation length, Rm is the average radius of the pore, and 2σ is the amplitude of corrugation.

Close modal

Similar to many other theoretical works on adsorption-induced deformation,15,19,20,22 we assume that only radial deformation of the pores results in the macroscopic deformation of the total sample. Using this assumption, we express the elastic Helmholtz free energy of the body Fel as a function of the order parameter Rm (see the  Appendix for details),

Fel(Rm)=F0+12KV0ζ2RmR0R02,
(2)

thus considering only the averaged radial deformation. Here, K is the effective bulk modulus of the sample, F0 is the reference Helmholtz free energy, V0 is the sample volume in the reference state, and R0 is the average pore radius in the reference state. The phenomenological parameter ζ determines how the strain on the microscopic scale translates to the macroscopic strain; it depends on the overall porosity of the sample, geometry and orientations of the pores, and elastic constants of the material. Let us consider the system at fixed volume and temperature, consisting of a porous body as an open system that is in contact with the reservoir with fixed chemical potential μb of fluid. Thus, the following thermodynamic potential has a minimum at equilibrium:

Φ=Fel+Ω+pV,
(3)

where Ω is the grand thermodynamic potential (GTP) of the adsorbed fluid, p is the pressure in the reservoir, and V is the volume of the sample. The GTP could also depend on additional parameters that are not defined by the chemical potential (the position of the liquid–vapor interface, for example). Minimizing Eq. (3) with respect to Rm as well as to other order parameters and approximating the GTP derivatives by their values at Rm = R0, we obtain

δVV0=ϕRmKζπlc(Rm2+σ2)dΩdRmpK,
(4)

where ϕ is the porosity of the sample. Here, we have expressed the reference sample volume as V0=mπlc(Rm2+σ2)/ϕ, where m is the total number of repeating sections in the pores. The volumetric pore load modulus M = /2ϕ can be defined as an elastic constant, determining the volumetric expansion of the materials under the solvation pressure.34,35 We can also introduce the parameter K/M=2ϕ/ζ, which is similar to the Biot coefficient.36,37 We will use ζ = 2ϕ in our calculations, which corresponds to the case of an incompressible solid. In this case, both elastic parameters K and M coincide. Because we neglect the effect of the coupling between deformation and adsorption, this approximation should be valid at sufficiently low strain magnitudes. In Eq. (3), Ω is the grand thermodynamic potential of the unique part of the pore. In this approximation, the additional order parameters are defined by the minimization of the GTP at the reference state of the solid. The solvation pressure for the corrugated pore can be defined as

f=Rm2πlc(Rm2+σ2)dΩdRm.
(5)

Thus, in order to estimate the solvation pressure, we need to calculate the GTP at fixed chemical potential and pressure. In the case of negligible corrugations, Eq. (5) matches the previously obtained equation for solvation pressure in cylindrical pores.

The GTP, in turn, is defined as a combination of the following contributions:

Ω=Ωfilm+Ωvapor+Sγlv,
(6)

where Ωfilm is the GTP of a liquid-like film on the surface, Ωvapor is the contribution due to the remaining vapor, and lv is the excess contribution associated with the liquid–vapor interface in the pore. Here, we omitted the additional excess contribution to the liquid–solid surface energy (γsl) arising from the over-account of fluid–fluid interactions (as the local model assumes the fluid structure to be the same as the bulk one). However, it should be almost constant since the relatively thick adsorbed film is formed. Thus, this contribution will result in a constant shift to the total solvation pressure. In addition, its magnitude should be smaller than the contribution from the solid–fluid interactions.

We will approximate both Ωvapor and Ωliquid in the local density approximation. The local formulation of density functional theory leads to the following GTP:

Ωρr=fT,ρrμbVextrρrdr,
(7)

where fT,ρr is the density of an intrinsic Helmholtz free energy that is approximated by a function of local number density ρr and Vextr is an external potential. The density distribution is defined by the minimization of Ω,

δΩ[ρ]δρr=0,
(8)

which is equivalent to μintr=μbVextr and defines the density profile in the system. Thus, the grand thermodynamic potential at equilibrium [Eq. (7)] can be written in the following equivalent form:

Ω=drP(T,μintr),
(9)

which can be interpreted as if the GTP density at each point is equal to the local pressure of fluid molecules as a function of the intrinsic chemical potential μint. Note that a similar approach was applied to the description of adsorption in pores38 and for the description of electric double layers in electrolyte solutions.39–41 

Moreover, the external potential now is treated as in Ref. 32 depending only on the distance of the tangent line to the surface. We use the simple form of the external potential, which simplifies the forthcoming calculations,

Vext(z,r)=wet/λ0,
(10)

where

t=Rp(z)r1+dRp/dz2
(11)

and λ0 is the effective radius of interaction.

Gas adsorption can cause the liquid bridge formation in the corrugated pores and that will affect the solvation pressure as well. We have demonstrated that for gas adsorption in model CMK-3 materials—porous solids composed of hexagonal arrangements of rods.25 Again, similar to the modeling adsorption in CMK-3 materials,42 we will consider three possible GTPs corresponding to three different filling regimes: multi-layer film, liquid bridge, and completely filled pore. The equilibrium transition between them will be denoted as desorption branch. The adsorption branch is treated as a metastable one. Thus, the subsequent regime cannot occur if the previous one may exist, i.e., the local minimum defining the previous regime must vanish.

We consider nitrogen at two different temperatures of 77 K and 154 K as examples for the calculations. We define the parameters of solid fluid interaction in the standard way, i.e., by the fitting of the reference non-porous isotherm measured on LiChrospher Si-1000 silica.43 Our adsorption model reduces to the well-known equation, describing liquid film growth on a planar surface, in the case of Rm and σ = 0. Thus, we fitted the reference isotherm by using the two-parametric equation defining the adsorbed film thickness,44 

t=λ0ln(kBTln(p/p0)/w),
(12)

presented here in the form used in the fitting procedure. We summarized all parameters used in the numerical calculations in Table I. Note that Eq. (12) is convenient for predicting adsorption-induced deformation because unlike others, such as Frenkel–Halsey–Hill isotherms, it does not give a divergence at the zero film thickness.22,24,45

TABLE I.

Solid–fluid parameters of interaction, nitrogen parameters for 77 K from NIST,46 and van der Waals parameters for nitrogen evaluated from the critical temperature and pressure.47 

Solid-fluid parametersFluid parameters: N2 77 KFluid parameters: N2 154 K
w/kB (K)λ0 (nm)γ (mN/m)P0 (MPa)ρl (mol/m3)aVdW (K nm3)bVdW (nm3)
−1038.5 0.28 8.88 0.097 152 28 832 273.4 0.064 24 
Solid-fluid parametersFluid parameters: N2 77 KFluid parameters: N2 154 K
w/kB (K)λ0 (nm)γ (mN/m)P0 (MPa)ρl (mol/m3)aVdW (K nm3)bVdW (nm3)
−1038.5 0.28 8.88 0.097 152 28 832 273.4 0.064 24 

In the case of N2 at 77 K, we use incompressible fluid approximation for pressure,

P(T,μint)p0+ρl(μintμ0),
(13)

where μ0, p0 = P(T, μ0), and ρl = ∂p0/∂μ0 are the chemical potential, pressure, and density of the liquid at saturation, respectively. In addition, we use the ideal gas approximation for the fluid in the reservoir and in the gas phase. Moreover, we assume that the properties in the latter are equal to the bulk properties due to the low values of the external potential. Thus, μint = μ0Vext(z, r) + kBT ln(p/p0) and p is the actual pressure value in the reservoir.

In the remaining part of this section, we will describe GTPs for different adsorption stages at 77 K and 154 K. In the case of bridged and film adsorption stages, we use specific approximations for the liquid–vapor interface based on the geometrical considerations.

The GTP in the case of a liquid film on the surface can be written as

Ωfilm=2πρlw0lcdzλ(z)Rp(z)λ(z)(rf(z)λ(z))erf(z)Rp(z)λ(z)0lcdzπrf2(z)p+π(p0+ρlkBTln(p/p0))(Rp2(z)rf2(z))+2πγlv0lcdzrf(z)1+drf(z)dz2,
(14)

where the last term takes into account the excess liquid–vapor interface contribution. Here, we also introduced the auxiliary function λ(z)=λ01+dR0/dz2. The form of the liquid–vapor interface is approximated by rf(z),

rf(z)=b+d2cos2πzlc.
(15)

This function repeats the form of the pore surface and, when d is negligible, corresponds to the cylindrical surface with radius b. Thus, by this function, we can cover a broad variety of possible interfaces. In the film regime, we use b and d as the order parameters of minimization. Minimizing the thermodynamic potential with respect to them, we can calculate the adsorbed amount

Nads=πρl0lcdz(Rp2(x)rf2(z))=Nfillπρllc(b2+d2)
(16)

and solvation pressure

ffilm=ρlwRmlc(Rm2+σ2)0lcdzλ(z)+(rf(z)λ(z))erf(z)Rp(z)λ(z)
(17)
+RmRm2+σ2p0+ρlkBTln(p/p0)
(18)

as functions of these two order parameters.

Instead of considering the bridged regime, we will start from the filled pore. This allows us to easily formulate both GTP and solvation pressure for the pore with liquid bridges. The grand thermodynamic potential can be written as

Ωfull=0lcdz(p0+ρlkBTln(p/p0))πRp2(z)+2πρlwλ(z)[Rp(z)λ(z)+λ(z)eRp(z)/λ(z)],
(19)

and the adsorbed amount per pore length is Nfill=πρllc(Rm2+σ2). In the case when the pore is completely filled with liquid, there are no order parameters; thus, additional minimization is not required. The solvation pressure for the filled regime is obtained by substitution of Eq. (19) into Eq. (5),

ffull=Rmlc(Rm2+σ2)0lcRp(z)(p0+ρlkBTln(p/p0))wρlλ(z)×1eRp(z)λ(z)dz,
(20)

and the typical dependence of solvation pressure on ln(p/p0) is obtained.

We approximate the gas bubble with the prolate spheroid cavity (a and c are the semi-axes and c > a). The grand thermodynamic potential can be written, using Eq. (19), as

Ωb=Ωfull+4π3a2cp0p+ρlkBTln(p/p0)+Slvγlv4πρlw×0cdzλ(z)(rb(z)λ(z))erb(z)Rp(z)λ(z)+λ(z)eRp(z)λ(z).
(21)

Slv is the surface area of the prolate spheroid,

Slv=2πa21+caεarcsin(ε),
(22)

where ε2 = 1 − a2/c2 is the eccentricity, and rb(z)=(a/c)c2z2. Thus, based on Eq. (19) and complementing it by the additional terms, taking into account the surface excess energy and over-account of the volume contributions, we formulate the GTP for the bridged regime. Using the semi-axis as an order parameter and minimizing Eq. (21) with respect to them, we can find a stable/metastable state of the system. The adsorbed amount in that regime is

Nbridge=Nfill4π3ρla2c.
(23)

The solvation pressure is also defined through ffull,

fb=ffull+2ρlwRmlc(Rm2+σ2)×0cdz(λ(z)rb(z))erb(z)Rp(z)λ(z)λ(z)eRp(z)λ(z).
(24)

The difference between solvation pressure in the filled and bridged regimes can be easily obtained from Eq. (24).

It is instructive to study the effect of high temperatures on the solvation pressure. Here, we will consider a temperature of 154 K, well above the critical temperature of bulk nitrogen equal to 126 K. In the case of the supercritical fluid, there is no liquid–vapor separation in the bulk fluid. Similarly, we assume that the grand thermodynamic potential in the pore can be approximated solely by Eq. (9). The form of solid–fluid interaction potential and values of interaction parameters remain unaltered. We use the van der Waals equation of state in order to take fluid compressibility into account.47 The choice of the model is made due to its simplicity and good recognition in the literature. The van der Waals parameters of interactions are given in Table I. The adsorbed amount

Nsc=2π0lcdz0Rp(z)drr(ρ(z,r)ρb)
(25)

and solvation pressure

f=Rmlc(Rm2+σ2)0lcdzRp(z)P(μ)r=Rp(z)+w0Rp(z)drrρ(z,r)λ(z)eRp(z)rλ(z)
(26)

are defined through the fluid density profile ρ(z, r) in the pore.

The minimization of the GTPs, defined in Secs. II AII C, was performed via the Matlab function fminsearch. The values of the order parameters defined by the minimum were further used in the calculations of solvation pressure and adsorbed amount. In all calculations, the mean radius of the pore and corrugation parameter were fixed to Rm = 3.6 nm and σ = 0.84 nm, respectively. In addition, we will compare our results for the corrugated pore with the results for the cylindrical pore with radius equal to Rm.

Figure 2 demonstrates the phase diagram of liquid bridges at 77 K. At different pairs (σ, lc) of microscopic corrugation parameters, the liquid bridge can form on the adsorption branch, on the desorption branch, or on both branches simultaneously. The resulting pressure isotherms are shown in Fig. 3, namely, the “iso-σ” line (0.84 nm) was calculated at four different corrugation lengths. The adsorption isotherms are shown alongside with solvation pressure isotherms, and the general hysteresis patterns are the same. The calculations were performed for four different values of lc – 11 nm, 14 nm, 28 nm, and 140 nm. At the lowest corrugation length, there is no bridge transition neither on the adsorption branch nor on the desorption branch. However, at higher lc values, the bridge transition appears on both branches. A further increase leads to the formation of two hysteresis loops. The solvation pressure repeats the adsorption isotherm behavior and has a discontinuity during every phase change in the pore. The solvation pressure in the bridge phase increases with an increase in relative pressure at first three corrugation lengths. However, in the case of lc = 140 nm, it demonstrates a non-monotonic behavior. In addition, the bridge formation decreases the magnitudes of the most pronounced steps on the desorption branch in comparison with the ideal cylindrical pore.

FIG. 2.

Phase diagram (σlc) calculated for the corrugated pore at 77 K. “D” defines the region of the parameters with the liquid bridge on the desorption branch and “A + D”—on both adsorption and desorption branches.

FIG. 2.

Phase diagram (σlc) calculated for the corrugated pore at 77 K. “D” defines the region of the parameters with the liquid bridge on the desorption branch and “A + D”—on both adsorption and desorption branches.

Close modal
FIG. 3.

Adsorbed amount and solvation pressure for N2 at 77 K in cylindrical and corrugated pores at different corrugation lengths from top to bottom: 11 nm, 14 nm, 28 nm, 140 nm. The corrugation parameter σ is fixed and equal to 0.84 nm. The absolute adsorbed amount is normalized to the corrugation length in each case. (Inset) The region with non-monotonic behavior of solvation pressure.

FIG. 3.

Adsorbed amount and solvation pressure for N2 at 77 K in cylindrical and corrugated pores at different corrugation lengths from top to bottom: 11 nm, 14 nm, 28 nm, 140 nm. The corrugation parameter σ is fixed and equal to 0.84 nm. The absolute adsorbed amount is normalized to the corrugation length in each case. (Inset) The region with non-monotonic behavior of solvation pressure.

Close modal

Eventually, the temperature increase should suppress the bridge transition in the corrugated pore. In order to study it, we calculated the excess adsorption isotherm and solvation pressure isotherms at twice higher temperature, 154 K. Nitrogen is above its bulk critical point at this temperature, and we have to take into account the effects of fluid compressibility. As described in Sec. II, we did it using the local density approximation and the van der Waals equation of state. The excess adsorbed amount (see Fig. 4) demonstrates the pronounced maximum at p/p0 ≈ 0.1. The solvation pressure isotherm has no discontinuities due to the absence of capillary condensation/evaporation transitions. The solvation pressure curve resembles the shape corresponding to some microporous materials.48 It should be noted that the non-monotonic behavior of solvation pressure in the corrugated pore may be an artifact of ζ choice. This could be because Fig. 4 represents the excess solvation pressure (f–p) and ζ affects only the first term, f.

FIG. 4.

Excess adsorbed amount per corrugation length and solvation pressure with respect to the bulk pressure of nitrogen at 154 K. The x axis is the pressure, calculated through the van der Waals equation of state.

FIG. 4.

Excess adsorbed amount per corrugation length and solvation pressure with respect to the bulk pressure of nitrogen at 154 K. The x axis is the pressure, calculated through the van der Waals equation of state.

Close modal

In this work, we formulated the model of adsorption-induced deformation of the material with corrugated cylindrical pores. The solvation pressure in corrugated pores showed a qualitatively different behavior in comparison with ideal cylindrical pores for nitrogen at 77 K. Namely, the complex hysteresis behavior observed in adsorption isotherms passes onto the solvation pressure isotherms. Additionally, the total magnitudes of solvation pressure jumps during the adsorption regime changes (film, bridged, or filled) are smaller in comparison with those of the ideal cylindrical pore. Thus, it may lead to an improved mechanical stability of the porous material during numerous adsorption–desorption cycles. We also believe that the measurements of adsorption-induced deformation could be used as an additional tool for porous material characterization because these signatures can be seen on solvation pressure and therefore on strain isotherms.

It is worth noting that the two-stage isotherms similar to our theoretical isotherms in Fig. 3 have been already observed experimentally11–13 and in computer simulations.12 In Ref. 13, the series of porous silicon samples were studied. The authors observed the two-stage adsorption process in open and closed pores with axially changing diameter. In Ref. 11, benzene adsorption on two mesoporous silicon samples with structured nano-channels was measured. The first sample, having the pores composed of two segments with different width, demonstrates the two-stage adsorption and desorption process. However, the second sample, which includes five periodic sections of different width, showing one-stage broad hysteresis. In addition, using the IR spectroscopy, the authors observed the cavitation mechanism of desorption in the five-channel sample. Adsorption of argon on porous anodic aluminum oxide duplex layers has two hysteresis loops, which is also confirmed by Monte Carlo simulations.12 

Despite some similarities, the experimental adsorption and strain isotherms do not demonstrate the abrupt behavior obtained within our model. Unfortunately, to the best of our knowledge, there are no experimental measurements of adsorption-induced deformation on the materials similar to that described above. From already existing data, measurements on SBA-15 would be the best candidates to demonstrate the predicted behavior. However, based on Fig. (2), one needs to use a sample with specific values of lc and σ (those also depend on the chosen fluid) to obtain the double step-wise behavior of the isotherm. At least two sets of experimental measurements on SBA-1534,45 do not demonstrate the desired transitions. Thus, probably, the most visible effect of corrugations is the narrowing of the hysteresis [see Fig. (1) and the case lc = 11 nm]. It was pointed by Gommes32 that accounting for the corrugations leads to better agreement with adsorption experimental data. In addition, we can expect similar improvements in model-experiment agreement in the case of strain measurement analysis. In principle, the bridges can be formed in more disordered materials, such as Vycor glass, due to correlated irregularities of the complex form. However, in this case, literature data50 also do not demonstrate the complex form of the adsorption isotherm. It can be due to several reasons, and the pore size distribution (PSD) is one of them. In order to estimate the influence of PSD on the isotherms, we perform the averaging22,51,52 by means of binomial distribution,

X=iwiXi,
(27)

where X is an averaging property and wi are the weighed coefficients, representing the fraction of pores with a certain value of Rm. Indeed, this procedure is not unique (for example, σ can also be varied) and cannot guarantee the correct reproduction of the experimental data. However, it can provide a possible qualitative explanation for the absence of the “zig-zags” on the deformation isotherms measured on the materials with the non-ideal polydispersed pores. The adsorption isotherms and solvation pressure averaged with a PSD using Eq. (27) are shown in Fig. 5. The remaining kink in Fig. 5 (right) could also be averaged in the real material.

FIG. 5.

Averaged excess adsorbed amount per corrugation length and solvation pressure at 77 K. Averaging is performed with symmetric binomial distribution having the maximum at R = 3.584 nm and covering the pore sizes from 3.08 nm to 4.06 nm. The corrugation length is 23.8 nm, and the corrugation parameter is 0.504 nm. The chosen values are similar to that proposed for SBA-15.32,49

FIG. 5.

Averaged excess adsorbed amount per corrugation length and solvation pressure at 77 K. Averaging is performed with symmetric binomial distribution having the maximum at R = 3.584 nm and covering the pore sizes from 3.08 nm to 4.06 nm. The corrugation length is 23.8 nm, and the corrugation parameter is 0.504 nm. The chosen values are similar to that proposed for SBA-15.32,49

Close modal

One of the interesting features is the continuous non-monotonic behavior of solvation pressure at significantly high corrugation lengths. For instance, it is demonstrated in Fig. 3 for lc = 140 nm. The maximum occurs at p/p0 ≈ 0.7, which corresponds to the beginning of the abrupt increase in the adsorbed amount. Taking the derivative of Eq. (24) with respect to the relative pressure, we obtain the following inequality:

lcRmkBTp/p02w0cdzrp(z)λ(z)rp(z)p/p0erp(z)Rp(z)λ(z)0,
(28)

defining the condition of a solvation pressure decrease. We treated the semi-axis of the spheroid as functions of relative pressure. This inequality can be simplified in the limit of big corrugation lengths, where λ(z) ≈ λ0 and |∂a/∂p| ≪ |∂c/∂p|. In addition, applying the same approximation to the adsorbed amount, one can obtain the following inequality:

Nbridgep>2πρlλ0lcRmkBTc33|w|pI,
(29)

where I denotes the integral from Eq. (28). This inequality suggests that the possibility of the solvation pressure decreasing depends not only on the adsorbed amount but also on the shape of the adsorption isotherm.

Our calculated adsorption isotherms and phase diagrams are very similar to the results from Ref. 32. It should be noted that our method is based on the approximation of the liquid–vapor interface by the elementary functions. Thus, the accuracy of the method depends on the quality of the interface parameterization. In addition, the deviations of the approximated liquid–vapor interface from the “real” solution could lead to inconsistency with the local Young–Laplace equation.

Up to now, we have discussed only the behavior of solvation pressure arising in corrugated pores during nitrogen adsorption at 77 K. The excess adsorption and solvation pressure isotherms calculated for supercritical nitrogen (scN2) are presented in Fig. 4. The adsorption isotherm has a well-known behavior with a pronounced maximum. The solvation pressure isotherm demonstrates the shape often observed for microporous materials, rather than for mesoporous ones. Nonetheless, the values of solvation pressure at 154 K are still significant; thus, scN2 could cause adsorption-induced deformation. Unlike adsorption at subcritical temperatures, the model does not predict any appreciable difference between adsorption isotherms and solvation pressures in corrugated and cylindrical pores. It can be explained that there are no fluid bridges in the corrugated pore due to almost negligible external potential on the axis of symmetry.

Finally, we would like to note that in Refs. 53 and 54, the authors studied the SBA-15 pore’s morphology by means of electron tomography and image analysis. The authors obtained the spatial homogeneity of the pore along the pore axis with the corrugation length of ≈5 nm. In addition, in Ref. 49, the authors observed the formation of periodical nano-bubbles in the SBA-15 pore. The obtained corrugation length was ≈11 nm. Such a discrepancy can be attributed to the differences in the synthesis procedure. The last value is close enough to the values of the corrugation length at which the bridging transitions could occur.

We proposed a model of adsorption-induced deformation of a material with corrugated cylindrical pores, representing, e.g., SBA-15 silica. Using the local density functional theory and linear elasticity theory, we studied the adsorption of nitrogen and solvation pressure in the pores, which is the driving force of adsorption-induced deformation. The results for subcritical adsorption (N2 at 77 K) revealed that complex hysteresis behavior, obtained previously for the adsorption isotherm,32 is also present on the solvation pressure isotherm. Namely, the solvation pressure in the presence of the liquid bridge is not monotonic in the pores with a large corrugation length. For the supercritical adsorption of nitrogen, the shape of the isotherms and solvation pressure curves becomes qualitatively different from that in the subcritical case, resembling those for microporous materials. Furthermore, the effects of the corrugation on isotherms and solvation pressure become negligible. These results will be useful in the analysis of the adsorption-induced deformation of materials with complex pore structures at sub- and supercritical temperatures.

This work was supported by a postdoc fellowship of the German Academic Exchange Service (DAAD). G.Y.G. acknowledges the support from the National Science Foundation (Grant No. CBET-1944495). The reported study was partially supported by the RFBR according to research Project No. 18-31-20015. The part regarding the adsorption of supercritical nitrogen was supported by the Russian Foundation for Basic Research under Grant No. 18-29-06008.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The total free energy of the dry sample consists of the bulk contribution and the excess contribution (arising from the presence of pore interfaces). We approximate the bulk contribution as a deformation of the smeared non-porous isotropic material with the same volume as the entire sample. The isotropic approximation should be reasonable for the disordered materials (such as glasses) on the scales, where all properties are averaged over the volume and there is no preferred direction. Moreover, to connect the strain in the effective matrix with the pore deformation, we will consider the strain as a function of the chosen order parameter.55 This approximation is reasonable for materials with a complex pore structure, where the actual strain tensor should be determined by the local pore geometry and their spatial arrangement in the sample. As we have restricted our considerations only to the volume compression/expansion, the strain tensor can be written in the following form:

uij=const×δij,
(A1)

where const is the function of the order parameter R. Thus, we assume that deformation is defined by the radial displacement of the average position of the corrugated pore wall. In the case of small deformations, the dimensionless strain tensor takes the form

uij=ζ3R0(RR0)δij,
(A2)

where ζ is the phenomenological parameter. In the case of an incompressible solid, ζ ≈ 2ϕ for both cylindrical and corrugated pores (for the latter in the case σlc only). However, in the general case, incompressibility of the solid skeleton is not assumed. The form of strain tensor reflects the idea that the change in the order parameter is the only origin of the macroscopic deformation.

The free energy density (with respect to the volume of the reference state) is56 

f=f0+ijσij0uij+12ijklSijkluijukl,
(A3)

where Sijlk is the stiffness tensor of the isotropic solid, which can be taken in the form Sijkl = λδijδkl + 3(Kλ)/2(δikδjl + δilδjk), with elasticity constants λ and K. In addition, we keep the linear term because the initial stress can be nonzero. Finally, we obtain the following expression:

f=f0+σ¯0ζR0(RR0)+12KζR02(RR0)2,
(A4)

where σ¯0 is the average initial stress in the material and K is the bulk modulus of the material. The total free energy of the solid consists of the volume and the surface contribution,

Fsolid=V0f+Fs(R),
(A5)

where Fs(R) is the surface free energy of the solid (here, we neglected the outer surface of the porous body) and V0 is the volume of the sample in the reference state. The reference state is chosen to be the state of the dry porous body, at which it has the initial stress caused by the existence of the excess surface energy. The surface free energy should be a function of the actual strain tensor in the porous body, which is why we consider it as a function the order parameter Rm, as the single parameter on which the strain in the smeared material depends. The exact dependence is not important for the present study, and in the limit of small deformation, Fs(R)Fs(R0)+dFs(R0)dR(RR0).

Let us consider the fixed volume system consisting of the material and some non-adsorbing gas at low pressure (heat carrier). At a fixed total volume of the system and temperature, the Helmholtz free energy is minimal. Due to the low pressure and absence of solid–fluid interactions, we can treat the solid state at the minimum as a reference state. Thus, neglecting the heat carrier pressure, the initial stress can be defined as

σ¯0=R0ζV0dFs(R0)dR.
(A6)

Thus, the linear terms in Eq. (A5) canceled out and the final equation appears as Hooke’s law. The resulting free energy can mimic the basic properties of the deformable solid but, however, remains only an approximation of the real free energy of the complex porous system under the mechanical load.

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