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.

N2 and CO2 sorption measurements with in situ dilatometry implemented in a commercial volumetric sorption instrument were performed at 77 and 273 K, respectively. The resolution of the linear deformation was about ±0.2 μm. To separate effects due to microporosity, external surface area and mesopores synthetic porous carbons (xerogels) with different external surface areas and microporosities were applied as a model system. The experimental data show that the relative length change of the monolithic carbon xerogels investigated passes different stages during ad- and desorption, which are connected to micropore-, multilayer- and mesopore-sorption. The length change observed in the range of micropore and surface adsorption was found to be nonmonotonic and to take negative as well as positive values, with the maximum swelling observed being on the order of 4‰. With respect to the length change, the micropore structure seems to have the most significant impact on the overall length change, while the external surface is only of minor importance. Quantiative analysis of the deformation according to the models of Bangham and Scherer for the length change in the range of multilayer- and mesopore-adsorption allows extracting the macrosopic as well as the skeletal Young's modulus.

We demonstrate that in-situ adsorption dilatometry provides a new opportunity for structural characterization of microporous carbons. We present experimental results for CO 2 adsorption at 293 K and in-situ deformation obtained by dilatometry on a synthetic monolithic carbon sample. The carbon deformation in the course of adsorption is non-monotonic: the strain isotherm shows the sample contraction at low adsorption followed by progressive expansion. To evaluate structural and mechanical properties of the sample from the experimental adsorption and strain isotherms, a kernel of theoretical adsorption isotherms is obtained with the grand canonical Monte Carlo simulation of CO 2 adsorption in a series of carbon micropores ranging from 0.22 to 2.0 nm. The respective kernel of adsorption stress isotherms is constructed using the thermodynamic model of adsorption stress. The pore volume and surface area distributions were calculated independently from a) the experimental excess adsorption isotherm by deconvoluting the generalized adsorption equation and b) the experimental strain isotherm by using the kernel of adsorption stress isotherms. The proposed method of determining the pore size distribution from the strain isotherm validates the thermodynamic model of adsorption stress in micropores and provides additional information about the sample material with respect to mechanical properties of the microporous matrix.

Adsorption-induced deformation of a monolithic, synthetic carbon of clearly distinguishable micro- and mesoporosity was analyzed by in situ dilatometry with N2 (77 K), Ar (77 K), CO2 (273 K), and H2O (298 K). A characteristic nonmonotonic shape of the strain isotherm showing contraction of the sample at initial micropore adsorption followed by expansion toward completion of micropore filling was found for all adsorbates. However, the extent of contraction and expansion varied significantly with the adsorbate type. The deformation differences observed were compared with the density ratio of the adsorbates within the micropores and the respective unconfined fluids. In particular, CO2 caused the least contraction of the sample, while in parallel adsorbed CO2 molecules were predicted to be considerably compacted inside carbon micropores compared to bulk liquid CO2. On the contrary, the packing of H2O molecules within carbon micropores is less dense than in the bulk liquid and adsorption of H2O produced the most pronounced contraction. N2 and Ar, both exhibiting essentially the same densities in adsorbed and bulk liquid phase, induced very similar deformation of the sample. These findings support theoretical predictions, which correlate adsorption-induced deformation and packing of molecules adsorbed in micropores. Additionally for the first time, we demonstrated with the N2 strain isotherm the existence of two nonmonotonic stages of subsequent contraction and expansion in the regions of micropore and mesopore filling. This characteristic behavior is expected for any micro- and mesoporous material.

Using the nonlocal density functional theory, we investigate adsorption of N2 (77 K), Ar (77 K), and CO2 (273 K) and respective adsorption-induced deformation of microporous carbons. We show that the smallest micropores comparable in size and even smaller than the nominal molecular diameter of the adsorbate contribute significantly to the development of the adsorption stress. While pores of approximately the nominal adsorbate diameter exhibit no adsorption stress regardless of their filling level, the smaller pores cause expansive adsorption stresses up to almost 4 GPa. Accounting for this effect, we determined the pore-size distribution of a synthetic microporous carbon by simultaneously fitting its experimental CO2 adsorption isotherm (273 K) and corresponding adsorption-induced strain measured by in situ dilatometry. Based on the pore-size distribution and the elastic modulus fitted from CO2 data, we predicted the sample's strain isotherms during N2 and Ar adsorption (77 K), which were found to be in reasonable agreement with respective experimental data. The comparison of calculations and experimental results suggests that adsorption-induced deformation caused by micropores is not limited to the low relative pressures typically associated with the micropore filling, but is effective over the whole relative pressure range up to saturation pressure.

Abstract:Experimental results on sorption-induced deformation during npentane desorption were obtained by in-situ dilatometry and in-situ small angle X-ray scattering (SAXS). The sample investigated was a silica-based monolith with hierarchical pore structure comprising a macroporous network of struts, each strut containing well-defined cylindrical mesopores ordered on a 2D hexagonal lattice. In-situ dilatometry and in-situ SAXS measurements revealed strain isotherms of similar shape, which are qualitatively in good agreement with recent theoretical predictions. From the relative pressure range of the liquid filled mesopores a pore load modulus of 1.5 GPa is determined. The relative pressure region of mono-and multilayer formation, however, reveals differences between the two independent methods. In particular, the net strain at saturation pressure is considerably larger for in-situ dilatometry. We attribute this observation to the different sensitivity of the two methods to anisotropic deformation in the hierarchical solid framework. While in-situ SAXS measures the mesopore lattice strain and is therefore exclusively sensitive to radial deformation of the struts, dilato-*Corresponding authors: Oskar Paris,

Mechanical properties of hierarchically structured nanoporous materials are determined by the solid phase stiffness and the pore network morphology. We analyze the mechanical stiffness of hierarchically structured silica monoliths synthesized via a sol–gel process, which possess a macroporous scaffold built of interconnected struts with hexagonally ordered cylindrical mesopores. We consider samples with and without microporosity within the mesopore walls and analyze them on the macroscopic level as well as on the microscopic level of the mesopores. Untreated as-prepared samples still containing some organic components and the respective calcined and sintered counterparts of varying microporosity are investigated. To determine Young’s moduli on the level of the macroscopic monoliths, we apply ultrasonic run time measurements, while Young’s moduli of the mesopore walls are obtained by analysis of the in situ strain isotherms during N2 adsorption at 77 K. For the latter, we extended our previously reported theoretical approach for this type of materials by incorporating the micropore effects, which are clearly not negligible in the calcined and most of the sintered samples. The comparison of the macro- and microscopic Young’s moduli reveals that both properties follow essentially the same trends, that is, calcination and sintering increase the mechanical stiffness on both levels. Consequently, stiffening of the monolithic samples can be primarily attributed to stiffening of the backbone material which is consistent with the fact that the morphology on the mesopore level is mainly preserved with the post-treatments applied.

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