Adsorption-Induced Deformation of Hierarchically Structured Mesoporous Silica—Effect of Pore-Level Anisotropy

The goal of this work is to understand adsorption-induced deformation of hierarchically structured porous silica exhibiting well-defined cylindrical mesopores. For this purpose, we performed an in situ dilatometry measurement on a calcined and sintered monolithic silica sample during the adsorption of N2 at 77 K. To analyze the experimental data, we extended the adsorption stress model to account for the anisotropy of cylindrical mesopores, i.e., we explicitly derived the adsorption stress tensor components in the axial and radial direction of the pore. For quantitative predictions of stresses and strains, we applied the theoretical framework of Derjaguin, Broekhoff, and de Boer for adsorption in mesopores and two mechanical models of silica rods with axially aligned pore channels: an idealized cylindrical tube model, which can be described analytically, and an ordered hexagonal array of cylindrical mesopores, whose mechanical response to adsorption stress was evaluated by 3D finite element calculations. The adsorption-induced strains predicted by both mechanical models are in good quantitative agreement making the cylindrical tube the preferable model for adsorption-induced strains due to its simple analytical nature. The theoretical results are compared with the in situ dilatometry data on a hierarchically structured silica monolith composed by a network of mesoporous struts of MCM-41 type morphology. Analyzing the experimental adsorption and strain data with the proposed theoretical framework, we find the adsorption-induced deformation of the monolithic sample being reasonably described by a superposition of axial and radial strains calculated on the mesopore level. The structural and mechanical parameters obtained from the model are in good agreement with expectations from independent measurements and literature, respectively.


Derivation of the Frumkin-Derjaguin equation for cylindrical pores
To derive the Frumkin-Derjaguin equation for a cylindrical pore of radius , length , surface area 2 and pore volume = 2 we consider the free energy of the adsorbate filled pore at saturation. According to Gibbs adsorption equation [1] we get: Here is the surface energy of the solid surface in vacuum and is the surface energy of the solid surface in contact with liquid. Using Eq. 6 and Eq. 9a we may write Combining Eq. S1 and S2 results in the Frumkin-Derjaguin equation for the cylindrical pore: In the limit of → ∞ Eq. 12 equals the common version of the Frumkin-Derjaguin equation.

Derivation of the Axial Stress in the Cylindrical Pore According to DBdB Theory
Within DBdB theory the grand potentials of adsorbed fluid in film and filled pore regime are given by [2]: Here the reference points are To calculate the axial (tangential) stress ,∥ we insert Eq. S3 and S4 into Eq. 5: The solution of the integral in Eq. S5a is obtained by integration by parts: Inserting Eq. 6, 9a and = 2 in Eq. S6 we get: Combining Eq. S5a and S7 results in Eq. 13a: Eq. S5b can be simplified by combination with Eq. 6 and 9b resulting in Eq. 13b:

Solution of the Lamé Problem for the Cylindrical Tube
When fluid is adsorbed inside the capillary tube, it exerts the adsorption stress on the inner wall and consequently causes strain within the tube. The resulting deformation is determined by the solution of the Lamé problem, here formulated in the cylindrical coordinates , and . The Lamé problem implies a system of equations for the radial, circumferential, and axial stresses ( , , ) and respective strains ( , , ). To simplify the situation we make several assumptions:  Due to the axial symmetry of the capillary tube, all stresses and strains are independent of the angle .
 The adsorbate density distribution does not depend on the axial coordinate . Consequently the same holds true for stresses and strains.
 The tube is sufficiently long ( ≫ ) to assume, that the axial stress is also independent of the spatial coordinate .
Based on these assumptions we obtain the following set of equations [3,4]: with the important consequence that Inserting Eq. S9, S10 and S11 into Eq. S8 the following equations are held for the strains: Eq. S12b and S12c correspond to Eq. 14 and 15.

Determination of the Reference Isotherm for DBdB Theory
For the determination of the disjoining pressure (ℎ) on sintered silica we prepared a purely macroporous reference sample. The reference sample was synthesized following the same protocols as the sample prepared for the in-situ dilatometry experiment [5,6], but dried in an autoclave using supercritical CO2 and subsequently calcined/sintered at 1000 °C for 20 min in ambient atmosphere. The bulk density of the monolithic sample = (0.647 ± 0.038) g/cm³ was determined after degassing at 110 °C for 1 d at gas pressures below 10 -3 mbar. The lack of micro-and mesopores was validated by scanning electron microscopy ( Figure S1) and N2 adsorption at 77 K ( Figure S2). From the N2 adsorption isotherm we derived the BET-surface area = (9.9 ± 0.5) m²/g [7], the disjoining pressure and the average film thickness ℎ by [8] = − ( 0 ⁄ ) ( 13) Here is the gas constant and = 34.66 cm³/mol the molar liquid volume of liquid nitrogen at the temperature of = 77.4 K. The resulting (ℎ) correlation is shown in Figure S3 along with an empirical fit (see e.g. [8]) The respective parameters applied were 1 = 178 MPa, 1 = 0.23 nm, 2 = 68 MPa and 2 = 0.073 nm. The reference isotherm resulting from the combination of Eq. 6 (in the limit of → ∞) and S15 is also shown in Figure S2.

Dependence of Axial and Circumferential Strain on the Poisson's Ratio
To investigate the impact of the nonporous backbone's Poisson's ratio on the theoretical strains we calculated the axial and radial stresses, ,⊥ and ,∥ , according to Eq. 10 and 13, respectively, for = 0.15, 0.2 and 0.25 (see Figure S4). This covers the full range of reasonable values for . Since the Young's modulus of the nonporous backbone is a simple scaling factor for the strains, we plotted strain multiplied by in Figure S4. As can be seen from Figure S4, the variation of does not change the shape of the strain isotherm, but results essentially in a scaling of ,⊥ and ,∥ similar to the Young's modulus, i.e. smaller values of lead to larger strains and vice versa. The overall variation of ,⊥ and ,∥ for different values of is rather small though.  Figure S5 shows a comparison of the experimental strain isotherm obtained for the model system via in-situ dilatometry and the respective modeling by the proposed theoretical framework (Eq. 20) as presented in Figure 8. Additionally, the prediction for the circumferential strain , (Eq. 14) corresponding to a hypothetical strain isotherm from in-situ scattering is included. All model parameters applied for the prediction of , are identical to the modeling of the in-situ dilatometry data. As can be seen from Figure S5, , exhibits a pronounced triangular behavior in the region of  p/p 0 capillary hysteresis that is typical for most of reported experimental data obtained by in-situ scattering on mesoporous materials, e.g. [9,10]. As expected, it is drastically different from the dilatometrically measured strain in the hysteresis region due to the anisotropic geometry of pore channels. Figure S5. Strain isotherm obtained by in-situ dilatometry for the model system complemented by the theoretical strain isotherms for in-situ dilatometry (Eq. 20) and the in-situ scattering (Eq. 14).