Physical Adsorption Characterization of Nanoporous · PDF filePhysical Adsorption Characterization of Nanoporous Materials Matthias Thommes During recent years, major progress has

Physical AdsorptionCharacterization ofNanoporous MaterialsMatthias Thommes

During recent years, major progress has been made in the understanding of the adsorption,

pore condensation and hysteresis behavior of fluids in novel ordered nanoporous materials

with well defined pore structure. This has led to major advances in the structural characteri-

zation by physical adsorption, also because of the development and availability of advanced

theoretical procedures based on statistical mechanics (e.g., density functional theory, mole-

cular simulation) which allows to describe adsorption and phase behavior of fluids in pores

on a molecular level. Very recent improvements allow even to take into account surface geo-

metrical in-homogeneity of the pore walls However, there are still many open questions

concerning the structural characterization of more complex porous systems. Important as-

pects of the major underlying mechanisms associated with the adsorption, pore condensa-

tion and hysteresis behavior of fluids in micro-mesoporous materials are reviewed and their

significance for advanced physical adsorption characterization is discussed.

Keywords: adsorption, nanoporous materials, pore condensation hysteresis

Received: March 22, 2010; accepted: April 13, 2010

1 Introduction

In recent years major progress has been madeconcerning the synthesis of highly ordered na-noporous materials (pore width range from2 – 50 nm) with tailored pore size and struc-ture, controlled surface functionality and theirapplications ([1 – 5] and references therein).Advances have also been made in the synthesisand structural characterization of micro-meso-porous materials such as mesoporous zeolites[6 – 9] and hierarchically organized pore struc-tures with an appropriate balance of micro-pores, mesopores and macropores, the latterbeing required to ensure the transport of thefluids to and from the smaller pores at a satis-factory rate. Recently, the synthesis of a novelclass of alumina/silica transition metal basedmaterials has been reported, which have par-tially pores between 1 and 2 nm, i.e, these no-vel materials bridge between zeolites andM41S materials [12]. An important new emer-ging class of solid state materials are metal-or-ganic framework materials (MOFs), which of-fer a wide range of potential applications (e.g.,gas storage, separation, catalysis, drug deliv-ery) [13 – 16].

A comprehensive characterization of theseporous materials with regard to pore size, sur-face area, porosity and pore size distribution isrequired in order to select and optimize theperformance of nanoporous and hierarchicallystructured materials in many industrial appli-cations [17 – 20].

In particular during the last decade, signifi-cant progress has been achieved in materialscharacterization and practical utilization be-cause of major improvements in the under-standing of the underlying mechanisms of ad-sorption in highly ordered mesoporousmaterials with simple geometries of known poresize (e.g., M41S materials) and consequently, inelaborating the theoretical foundations of ad-sorption characterization [22 – 24]. This has leadto the development of microscopic approachessuch as the nonlocal density functional theory(NLDFT) and methods based on molecularsimulation (e.g., Grand Canonical Monte Carlosimulation) which allow to describe adsorptionand phase behavior of fluids in pores on a mole-cular level [25 – 30]. It has been demonstratedthat the application of these novel theoreticaland molecular simulation based methods leadsto: (i) a much more accurate pore size analysis

An important newemerging class ofsolid state materialsare metal-organicframework mate-rials.

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DOI: 10.1002/cite.201000064

[31, 32, 23], and (ii) allows performing pore sizeanalysis over the complete micro/mesopore sizerange [e.g., 9, 21, 22, 32]. Appropriate methodsfor pore size analysis based on NLDFT and mo-lecular simulation are meanwhile commerciallyavailable for many important adsorptive/adsor-bent systems. This includes hybrid methodsthat assume various pore geometries for the mi-cro- and mesopore size range, as it can be foundfor materials with hierarchical pore structures.The application of the NLDFT for micro-and me-sopore size analysis has also been featured inISO (International standard organization, ISO)standards [33].

These advances have been accompanied bythe progress made in the development of var-ious experimental techniques, such as gas ad-sorption, X-ray diffraction (XRD), small angle x-ray and neutron scattering (SAXS and SANS),mercury porosimetry, electron microscopy(scanning and transmission), thermoporome-try, NMR-methods, and others [34 – 46]. In orderto explore details of the adsorption mechanismand phase behavior of fluids in more complexporous systems (e.g., micro-mesoporous zeo-lites, hierarchically structured porous materi-als), it is advantageous to combine various ex-perimental methods (e.g., coupling adsorptionexperiments with SAXS and SANS, i.e, in-situ-scattering [37, 42, 44 – 46]. However, among allthese methods, gas adsorption is still the mostpopular one because it allows assessing a widerange of pore sizes, covering essentially thecompleted micro-and mesopore range. Further-more, gas adsorption techniques are convenientto use and are less cost-intensive than some ofthe other methods. In recent years, automatedadsorption equipment has been installed in al-most every organization concerned with thesynthesis and characterization of nanoporousmaterials. The development of commercial ad-sorption equipment has been accompanied bythe installation of user-friendly data reductionsoftware; nevertheless it is crucial to understandthe fundamental principles involved in the inter-pretation of the isotherm data in order to arriveat a meaningful surface area and pore size analy-sis. In this paper focus is on some selected, im-portant aspects of surface area and pore sizeanalysis in particular in light of the progressmade in this area over the last decade or so.

2 Physical Adsorption in Nano-pores

2.1 General Aspects

Physisorption (physical adsorption) occurswhenever a gas (the adsorptive) is brought into

contact with the surface of a solid (the adsor-bent). The matter in the adsorbed state isknown as the adsorbate, as distinct from theadsorptive, which is the gas or vapor to be ad-sorbed. The forces involved in physisorptionare the van-der Waals forces and always in-clude the long-range London dispersion forcesand the short-range intermolecular repulsion.These combined forces give rise to nonspecificmolecular interactions. Specific interactionscome into play when polar molecules are ad-sorbed on ionic or polar surfaces but, as longas there is no form of chemical bonding, theprocess is still regarded as physisorption. Phy-sical adsorption processes in porous materialsis governed by the interplay between thestrength of fluid-wall and fluid-fluid interac-tions as well as the effects of confined porespace on the state and thermodynamic stabilityof fluids confined to narrow pores. This is re-flected in the shape or type of the adsorptionisotherm. Within this context the InternationalUnion of Pure and Applied Chemistry (IU-PAC) has published a classification of six typesof adsorption isotherms [17] and proposed toclassify pores by their internal pore width. Thepore width is defined as the diameter in caseof a cylindrical pore and as the distance be-tween opposite walls in case of a slit pore), i.e.,Micropore: pore of internal width less than2 nm; Mesopore: pore of internal width be-tween 2 and 50 nm; Macropore: pore of inter-nal width greater than 50 nm. The microporerange is subdivided into those smaller thanabout 0.7 nm (ultramicropores) and those inthe range from 0.7 – 2 nm (supermicropores).The pore size is generally specified as the inter-nal pore width (for slit-like pores) pore radius/diameter (for cylindrical and spherical pores).It is important to note that the internal or ef-fective pore width differs from the distance be-tween the centers of surface atoms, which isusually employed in simulation work (i.e., theouter atoms of solid in the opposite walls of apore). It has become popular to refer to micro-pores and mesopores as nanopores. The gasadsorption technique allows of course only todetermine the volume of open pores. Closedporosity cannot be accessed, but can be derivedif the true density and particle(bulk) density ofthe materials are known. Porosity is defined asthe ratio of the volume of pores and voids tothe volume occupied by the solid. Further, itshould be noted that it is not always easy todistinguish between roughness and porosity.In principle, a simple convention is to refer toa solid as porous if the surface irregularitiesare deeper than they are wide.

The adsorbed amount as a function of pres-sure (or relative pressure P/P0, where P0 is the

Appropriate meth-ods for pore sizeanalysis based onNLDFTand molecu-lar simulation aremeanwhile commer-cially available.

The van-der Waalsforces and alwaysinclude the long-range Londondispersion forcesand the short-rangeintermolecular re-pulsion.

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saturation pressure of the adsorptive at a giventemperature) can be measured by volumetric(manometric) and gravimetric methods, car-rier gas and calorimetric techniques, nuclearresonance as well as by a combination of ca-lorimetric and impedance spectroscopic mea-surements [34 – 47]. However, the most fre-quently used methods are the volumetric(manometric) and the gravimetric methods.The gravimetric method is convenient to usefor the study of vapor adsorption not too farfrom room temperature, whereas the volu-metric (manometric) method has advantagesfor the measurement of nitrogen, argon andkrypton adsorption at cryogenic temperatures(77.4 K and 87.3 K), which are mainly used forsurface area and pore size characterization [17].Details concerning manometric (volumetric)and gravimetric experimental adsorption tech-niques can be found in [18 – 20, 48 – 52].

Nitrogen at 77 K is considered to be a stan-dard adsorptive for surface area and pore sizeanalysis, but it is meanwhile generally acceptedthat nitrogen adsorption is not satisfactory withregard to a quantitative assessment of the mi-croporosity, especially in the range of ultrami-cropores (pore widths < 0.7 nm). Consequently,alternative probe molecules have been sug-gested, e.g., argon and carbon dioxide. Formany microporous systems (in particular zeo-lites) the use of argon as adsorptive at its boilingtemperature (87.3 K) appears to be very useful[9, 20 – 22, 24, 54]. When compared to nitrogenand carbon dioxide, it exhibits weaker attractivefluid-pore wall attractions for most adsorbents,which – during adsorption – does not give riseto specific interactions (like nitrogen and car-bon dioxide because of their quadrupole mo-ments) with most of surface functional groupsand exposed ions. As a consequence, for in-stance in case of zeolites, argon fills microporesof dimensions 0.5 – 1 nm at much higher rela-tive pressures (i.e., 10–5 < P/P0 < 10–3) thannitrogen (i.e., 10–7 < P/P0 < 10–5), which leadsto accelerated diffusion and equilibration pro-cesses, and allows to obtain accurate high reso-lution adsorption isotherms within a reason-able time frame [19 – 21, 50]. Because of thelack of specific interaction between argon andthe pore walls, the correlation between poresize and pore filling pressure is much morestraightforward for argon as compared to nitro-gen carbon dioxid. However, it has to be notedthat contrary to 87.3 K, argon adsorption at li-quid nitrogen temperature (77.4 K) is not thebest choice for pore size/porosity characteriza-tion because of various reasons asscociated withthe fact that at 77.4 K argon is ca. 6.5 K belowthe triple point temperature of bulk argon[22, 23, 20].

Despite the advantages which argon adsorp-tion at 87.3 K offers, pore filling of ultramicro-pores still occurs at very low pressures (i.e,. tur-bomolecular pump vacuum is needed).Associated with the low pressures is as indi-cated above, the well-known problem of re-stricted diffusion, which prevents nitrogenmolecules and also argon molecules from en-tering the narrowest micropores, i.e., pores ofwidths < ca. 0.45 nm. Alternatives for the deter-mination of the total pore volume are CO2 ad-sorption at room temperature. While CO2 ad-soption at 273 K is frequently used for theultramicopore analysis of carbonaceous mate-rials [53], it is not a good choice for the poresize analysis of materials with polar sites,mainly because of the very specific interactionsthat CO2 can have with functional groups onthe surface. However, it can still be used for as-sessing pore volume/porosity; the usefullnessof CO2 adsorption for the determining thepore volumes of NaX zeolites has been demon-strated (e.g, [54]).

Krypton adsorption at 77.4 K is more or lessexclusively used for low surface area analysisof materials such as thin films [20] althoughsome attempts to apply it for the pore size ana-lysis of thin films have been reported as well[e.g., 55]. If applied at 87.3 K, Krypton adsorp-tion also allows to obtain the pore size distribu-tion of thin mesoporous silica films with porediameters ranging from below 1 nm up to∼ 9 nm; although krypton at 87.3 K is ca. 30 Kbelow the bulk triple point temperature, if con-fined to cylindrical silica pores with diameter< 9 nm it appears to be in a supercooled liquidstate [56].

2.2 Adsorption Mechanism

The sorption behavior in micropores (porewidth < 2 nm) is dominated almost entirely bythe interactions between fluid molecules andthe pore walls; in fact the adsorption potentialsof the opposite pore walls are overlapping. As aconsequence micropores fill through a contin-uous process (i.e., no phase transition). Thefilling of the narrowest micropores (i.e., ofwidth equivalent to no more than two or threemolecular diameters) takes place at low relativepressures (at P/P0 < 0.01). This process hasbeen termed “primary micropore filling“ Fill-ing of the wider micropores may occur overa much wider range of relative pressure(P/P0 ≈ 0.01 – 0.2). The enhancement of the ad-sorbent-adsorbate interaction energy in thepore center is now very small and the in-creased adsorption is mainly due to coopera-tive adsorbate-adsorbate interactions.

The gravimetricmethod is con-venient to use forthe study of vaporadsorption not toofar from roomtemperature.

Restricted diffusionprevents nitrogenmolecules from en-tering the narrow-est micropores.

Micropores fillthrough a con-tinuous process.



In contrast, the sorption behavior in meso-pores depends not only on the fluid-wall attrac-tion, but also on the attractive interactions be-tween the fluid molecules. This leads to theoccurrence of multilayer adsorption and capil-lary (pore) condensation (at P/P0 > ≈ 0.2) thepore walls are covered by a multilayer adsorbedfilm at the onset of pore condensation. The sta-bility of the adsorbed multilayer film for in-stance in a cylindrical pore is determined bythe long-range van der Waals interactions, andby the surface tension and curvature of the li-quid-vapor interface [23, 58 – 60]. For smallfilm thickness the adsorption potential domi-nates. However, when the adsorbed film be-comes thicker, the adsorption potential be-comes less important, whereas surfacetension/curvature effects become significant.At a certain critical thickness tc, the multilayerfilm cannot be stabilized anymore, and porecondensation occurs in the core of the pore,controlled by intermolecular forces in the corefluid. Pore condensation represents a phenom-enon whereby gas condenses to a liquid-likephase in pores at a pressure less than the sa-turation pressure P0 of the bulk fluid. It repre-sents an example of a shifted bulk transitionunder the influence of the attractive fluid-wallinteractions. For pores of uniform shape andwidth (ideal slit-like or cylindrical mesopores)pore condensation can be classically describedon the basis of the Kelvin equation [61], i.e.,the shift of the gas-liquid phase transition of aconfined fluid from bulk coexistence, is ex-pressed in macroscopic quantities like the sur-face tension c of the bulk fluid, the densities ofthe coexistent liquid ql and gas qg (Dq = ql - qg)and the contact angle h of the liquid meniscusagainst the pore wall. For cylindrical pores themodified Kelvin equation [62] is given by:ln(P/P0) -2ccosh/RTDq(rp–tc), where R is theuniversal gas constant, rp the pore radius andtc the thickness of an adsorbed multilayer film,which is formed prior to pore condensation.The occurrence of pore condensation is ex-pected as long as the contact angle is below90°. A contact angle of 0° (i.e., complete wet-ting) is usually assumed in case of nitrogenand argon adsorption at 77.4 K and 87.3 K, re-spectively.

The Kelvin equation provides a relationshipbetween the pore diameter and the pore con-densation pressure, and predicts that pore con-densation shifts to a higher relative pressurewith increasing pore diameter and tempera-ture. Hence, the modified Kelvin equationserves as the basis for many methods appliedfor mesopore analysis, including the widelyused Barett-Joyner-Halenda method (BJH).However, the validity of macroscopic, thermo-

dynamic concepts such as the Kelvin equationand related methods becomes questionable fornarrow mesopores (i.e, pore diameter smallerthan ca. 15 nm (a comprehensive review is gi-ven in [23])

Capillary condensation is very often accom-panied by hysteresis (Fig. 1), which of courseintrodcues a considerable complication forpore size analysis, but if interpreted correctly,provides important information about the porestructure/network, which is crucial for obtain-ing a comprehensive and accurate textural ana-lysis of advanced nanoporous materials. Hys-teresis can be observed in single pores as wellas in pore networks [23, 67 – 71]. Generally,hysteresis is being considered: (i) on the levelof a single pore of a given shape, (ii) coopera-tive effects due to the specifics of connectivityof the pore network, and (iii) in highly disor-dered, and for inhomogenous porous materialsa combination of kinetic and thermodynamiceffects spanning the complete disordered poresystem has to be taken into account. Progresshas been achieved in understanding the under-lying internal dynamics of hysteresis in disor-dered pore systems [71], however a discussionof this topic is beyond the scope of this Sec-tion.

An empirical classification of hysteresisloops was given by IUPAC (Fig. 1), in whichthe shape of the hysteresis loops (types H1 –H4) are correlated with the texture of the ad-sorbent. According to this classification, typeH1 is often associated with porous materialsexhibiting a narrow distribution of relativelyuniform (cylindrical-like) pores. Materials thatgive rise to H2 hysteresis contain a more com-plex pore structure in which network effects(e.g., pore blocking/percolation) are important.

Isotherms with type H3 hysteresis do not ex-hibit any limiting adsorption at high P/P0. Thisbehavior can for instance be caused by the exis-tence of non-rigid aggregates of plate-like parti-cles or assemblages of slit-shaped pores and inprinciple should not be expected to provide areliable assessment of either the pore size dis-tribution or the total pore volume. H4 hyster-esis loops are generally observed with complexmaterials containing both micropores and me-sopores. Both, types H3 and H4 hysteresis con-tain a characteristic step-down in the desorptionbranch associated with the hysteresis loop clo-sure.

On the pore level or independent pore mod-el, adsorption hysteresis is considered as an in-trinsic property of the vapor-liquid phase tran-sition in a finite volume system. A classicalscenario of capillary condensation implies thatthe vapor-liquid transition is delayed due to theexistence of metastable adsorption films and

Hysteresis intro-dcues a consider-able complicationfor pore sizeanalysis.

Adsorption hyster-esis is considered asan intrinsic propertyof the vapor-liquidphase transition in afinite volumesystem.



hindered nucleation of liquid bridges [67 – 70].In open uniform cylindrical pores of finitelength, these metastabilities occur only on theadsorption branch. Indeed, in an open porefilled by liquid-like condensate, the liquid-va-por interface is already present, and evapora-tion occurs without nucleation, via a recedingmeniscus. That is (as indicated before), thedesorption process is associated with the equi-librium vapor-liquid transition, whereas hyster-esis is caused by the fact that condensation oc-curs delayed due to the metastabilitiesassociated with the nucleation of liquidbridges. Typically, a hysteresis loop of type H1is observed. Meanwhile, modern, microscopicapproaches such as non-local density func-tional theory (NLDFT) and molecular simu-lation (e.g, Grand Canonical Monte-Carlosimulation) are capable of qualitatively andquantitatively predicting the pore condensationand hysteresis behavior of fluids in orderedmesoporous materials . NLDFT correctly pre-dicts (i) the positions of equilibrium vapor-li-quid transition which is associated with thedesorption branch of the isotherm in a pore ofgiven size and geometry; (ii) the pressurewhere capillary condensation occurs by takinginto account delayed condensation due to themetastability associated with the nucleation ofliquid bridges (the resulting NLDFT method/kernel is based on so-called metastable adsorp-tion isotherms) [31]. Hence, if the hysteresis iscaused solely by the delayed condensation ef-fect, the pore sizes calculated from the adsorp-tion branch (by applying the NLDFT kernel ofmetastable adsorption isotherms) and deso-rption branch (by applying the NLDFT kernelof equilibrium isotherms) should be in agree-ment. This was indeed found for MCM-41,SBA 15 silicas [31, 23], which clearly confirmedthe applicability of the so-called single (or inde-pendent) pore model for these materials. An

example is shown in Fig. 2 which shows nitro-gen adsorption in SB 15 silica.

Hysteresis in pore networks is expected tobe more complex and very often hysteresisloops which reflect the shapes of types H2 toH4 are observed. However, some novel meso-porous materials such as MCM 48 and KIT 6silica, which consist of ordered 3D pore net-works can reveal perfect type H1 adsorptionhysteresis [69 – 74], which indicates that porechannels do not exhibit constrictions whichwould otherwise give rise to type H2 hysteresisdue to pore blocking/percolation effects andtherefore would lead to deviations from typeH1 hysteresis. However, it has been observedthat hysteresis loops associated with the porecondensation of fluids (e.g., nitrogen, argon)in ordered three dimensional pore systems of

0 0.2 0.4 0.6 0.8 1P/P0

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Figure 1. IUPAC classificationof hysteresis loops.

Figure 2. a) Nitrogen adsorp-tion/desorption at 77.35 K inSBA-15silica.b) NLDFT pore size distribu-tions from adsorption- (appli-cation of NLDFT metastableadsorption isotherm kernel)and desorption (applicationof NLDFT equilibrium iso-therms kernel).



MCM-48 or KIT-6 silica are generally more nar-row as compared to the hysteresis loops ob-served in the pseudo-one-dimensional poresystems of MCM-41 or certain SBA-15 silicamaterial [69, 71, 72]. This suggests that even inthe absence of typical network effects (e.g.,pore blocking) pore connectivity can have animpact on the width of the hysteresis loop, i.e,.it appears that that the interconnectivity ofpores may lead to so-called the initiated/facili-tated condensation [75, 78, 79, 80]. In porenetworks neighbouring pores with a slightlysmaller pore diameter are already filled withcondensate, which exhibits a meniscus, andthis interface can principally advance into lar-ger neighbouring pores. This could then re-duce the nucleation barriers associated withcapillary condensation (in these slightly largerneighbouring pore segments) and thereforethe pressure range over which metastablestates extend. On the other hand, it is not to beexpected (in the absence of pore blocking ef-fects) that the interconnectivity of pores couldhave any appreciable influence on the positionof the capillary evaporation. Hence, one shouldbe able to calculate a reliable pore size distribu-tion for ordered 3D systems from the deso-rption branch.

Hysteresis phenomena in pore networksconsisting of ink-bottle type are quite complex.Two basic mechanisms of desorption in porenetworks are distinquished as pore blockingpercolation and cavitation. The former me-chanism was introduced in the early studies ofcapillary hysteresis and, therefore is some-times called ink-bottle or classical pore blockingmechanism [77 – 82]. It is well understood thatevaporation the capillary condensate from anetwork of ink-bottle is obstructed by the pore

constrictions. In this case, desorption from thepore body may occur only after emptying of itsneck. In other words, desorption from the necktriggers evaporation in the blocked pore. Thus,the vapor pressure of desorption from the porebody depends on the neck size and networkconnectivity, as well as, on the state of theneighbouring pores. The onset of evaporationfrom the pore network is associated with thepercolation treshhold and the formation of acontinuous cluster of pores open to the exter-nal surface [80 – 86]. The percolation mechan-ism is observed in the pore networks withsufficiently large necks. Some typical porestructures where pore blocking is expected areshown in Fig. 3.

Conventional type H2 hysteresis will also oc-cur in the case of a wide distribution of inde-pendent pores with the same or similar necksize, or in a network where the necksize distri-bution is much more narrow than the sizedistribution of the main cavities (e.g., poreblocking/percolation phenomena play and im-portant role in porous vycor glass [35]). Re-cently, a different type of hysteresis loop,which looks somewhat like an inverse type H2hysteresis has been associated with the occur-rence of pore blocking as well (Fig. 3). In thiscase the desorption branch is less steep thanthe adsorption branch. Such hysteresis couldbe observed in materials where the pore sizedistribution of the main pores is more narrowthan the pore size distribution of the entran-ce(neck) diameters. Inverse type H2 hysteresishas been observed for instance in mesoporousfoam consisting of polyhedral foam cells of60 – 70 nm diameter, interconnected by cylind-rical access channels with several characteristicsizes for the latter [47], or in materials such asFDU-1 silica [91] or KIT-5 silica [92], where theentrances to the spherical pores had been wi-dened by either calcination and or hydrother-mal treatment, respectively. In this case, thedistribution of necks/constrictions is muchwider than the distribution of main pore cav-ities, therefore the adsorption/condensationbranch is much steeper than the desorptionbranch. Hence, the distribution of neck sizescan be obtained from an analysis of the deso-rption branch, whereas the pore/cavity size dis-tribution is only available from an analysis ofthe adsorption branch (e.g., by applying amethod for pore size analysis which correctlytakes into account the delay in condensationsuch as the NLDFT kernel of metastable ad-sorption isotherms; Fig. 2) .

Theoretical and experimental studies [32, 94]have revealed that if the neck diameter is smal-ler than a certain critical width (estimated tobe ca. 6 nm for nitrogen at 77.4 K), the me-Figure 3. Schematic illustration of hysteresis in ink-bottle pores.

It appears that thatthe interconnectiv-ity of pores maylead to so-called theinitiated/facilitatedcondensation.

Evaporation of thecapillary condensatefrom a network ofink-bottle is ob-structed by the poreconstrictions.



chanism of desorption from the pore body in-volves cavitation - spontaneous nucleation andgrowth of gas bubbles in the metastable con-densed fluid (Fig. 4). In this case, the porebody empties while the pore neck remainsfilled. It is important to note that if the poreneck is below the critical neck size, the actualwidth of the pore neck appears not to play anyrole for the pressure where cavitation occurs,i.e., the cavitation pressure depends essentiallyon the thermophysical properties of the fluidin the main pore cavity. Cavitation controlledevaporation can for instance be found in mate-rials such as SBA-16 [e.g., 99] and in silicaswith hierarchical pores structures such as KLE,and KLE/IL silica [32], mesoporous zeolites[e.g., 9] and some clays [e.g., 100]. As men-tioned before, it is expected that at a given tem-perature, the neck size controls whether poreblocking or cavitation occurs. Above a certaincritical neck size (≈ 6 nm) pore blocking occurs,and below this cavitation controlled evapora-tion takes place. Hence, by varying the necksize/entrances to the main pore system, oneshould be able to observe such a transitionfrom cavitation induced evaporation to poreblocking. Indeed such results have been re-ported for SBA-16 silica [99], FDU-1 silica [91],and KIT-5 silica [92]. On the other hand, thesame phenomena can be observed by varyingthe temperature of the adsorption experimentfor a given adsorbend with ink-bottle geometry.

In order to detect which mechanism isdominant, an adsorption test was suggested in[32]. This test is based on measuring adsorp-tion isotherms with different adsorbates (suchas nitrogen and argon) and/or at different tem-perature and comparing PSDs calculated fromthe data obtained at these different conditions.In the case of pore blocking, the pressure ofevaporation is controlled by the size of con-necting pores. Therefore, PSDs calculatedfrom the desorption branches should be inde-pendent of the choice of the adsorbate or tem-perature. This has indeed been found for in-stance in porous vycor glass which is known togive rise to pore blocking/percolation phenom-ena [32]. In the case of cavitation, the pressureof desorption depends on the adsorbate andtemperature and is not correlated with the sizeof connecting pores. Hence, PSDs calculatedfrom the desorption branch of the hysteresisloop are artificial; they do not reflect the realpore sizes and they should depend on the choiceof the adsorptive and/or temperature which hadbeen demonstrated for various silica materialswith hierarchical pore structures [32, 101].In addition to hierarchically structured materi-als (e.g. KLE/IL silica), and micro-mesoporouszeolites, plugged hexagonal templated silica

(PHTS) material with combined micro- andmesopores and a tunable amount of both openand inkbottle pores gained recently some at-tention [102, 103] Fig. 5 shows a schematic iso-therm typical for adorption in plugged SBA-15silica.

The adsorption/desorption isotherm is con-sistent with a structure which exists of bothopen and blocked cylindrical mesopores. Thetwo-sep desorption isotherm indicates the oc-currence of both equilibrum evaporation/deso-rption and pore blocking/cavitation effects.The desorption at higher pressures is asso-ciated with the evaporation of liquid from openpores. On the other hand, blocked mesoporesremain filled until they empty via cavitation.

Based on the discussed examples, it appearsthat cavitation induced evaporation appears tobe important for many micro/mesoporous so-lids and is responsible for the often observedcharacteristic step down in the desorption iso-therms associated with hysteresis loop closure(see type H3 and H4 hysteresis loops in theIUPAC classification (Fig. 1). In the past, thischaracteristic step down was discussed withinthe framework of the clasccial tensile strengthhypothesis [18, 104 – 106]. Here it was assumed

Figure 4. Schematic illustration of pore blocking and cavitation controlled evapora-tion. Adapted from [32].

Figure 5. Characteristic sorp-tion isotherm as it can befound in plugged cylindricalpores.

At a given tempera-ture, the neck sizecontrols whetherpore blocking orcavitation occurs.



that the tensile stress limit of condensed fluid(the pressure where cavitation induced eva-poration occurs) does not depend on the nat-ure and pore structure of the adsorbent, yet isa universal feature of the adsorptive. However,in contrast to this classical viewpoint, very re-cent work has clearly revealed that the onset ofcavitation (and the so-called lower closurepoint of hysteresis) depends on pore/cavitysize and pore geometry for for pore diameterssmaller than ≈ 11 nm, but remains practicallyunchanged for samples with large pores [101].

3 Surface Area and Pore SizeAnalysis

3.1 Application of the BET (Brunauer,Emmett and Teller) method

Surface area is a crucial parameter for optimiz-ing the use of porous materials in many appli-cations, and has also recently be discussed incontext with the novel class of Metal-Organic-Framework (MOF) materials, where extremelyhigh specific surface areas (> 3000 m2g–1) havebeen reported [e.g., 14]. Due to the complexnature of micro/mesoporous materials no sin-gle experimental technique can be expected toprovide an evaluation of the absolute surfacearea, however the most frequently appliedmethod is the BET method introcuded morethan sixty years ago by Brunauer, Emmett andTeller [103]. Usually, two stages are involved inthe evaluation of the BET area. First, it is ne-cessary to transform a physisorption isotherminto the ‘BET plot’ and from there to derive thevalue of the BET monolayer capacity, nm. Thesecond stage is the calculation of the specificsurface area, S, which requires knowledge ofthe molecular cross-sectional area. The mono-layer capacity nm is calculated from the adsorp-tion isotherm using the BET equation (Eq. (1))

1/[n((P0/P)-1)] = (1/nmC) + [(C-1)/nmC] (P/P0)(1)

where n is the adsorbed amount, nm is themonolayer capacity and C is an empirical con-stant which gives an indication of the order ofmagnitude of the attractive adsorbent-adsor-bate interactions. In the original work of Bru-nauer, Emmett and Tellerit was found that typeII nitrogen isotherms (according to the IUPACclassification [10]) on various nonporous adsor-bents gave linear BET plots over the approxi-mate range p/p° – 0.05 – 0.3. The specific sur-face area S can then be obtained from themonolayer capacity nm by the application ofthe simple equation: S = Nm Lr, where L is the

Avogadro constant and r is the so-called cross-sectional area (the average area occupied byeach molecule in a completed monolayer).

The BET equation is applicable for surfacearea analysis of nonporous- and mesoporousmaterials consisting of pores of wide pore dia-meter, but is in a strict sense not applicable formicroporous adsorbents (for a critical apprai-sal of the BET method is given in [19, 20]).Hence, the surface area obtained by applyingthe BET method on adsorption isotherms frommicroporous materials reflects as a kind of ap-parent or equivalent BET area [19]. A problemdirectly related to the discussion concerningthe applicability of the BET method for asses-sing the surface areas of microporous materi-als is the determination of the proper relativepressure range for applying the BET method.If the BET equation is applied within it’s classi-cal range (rel. pressure range 0.05 – 0.3) on ad-sorption data obtained on microporous materi-als, one does very often not find a linear range,the C-constant maybe negative (which is un-physical) and the obtained BET area dependson the selected data points. Recently a proce-dure was suggested which allows to determinethis linear BET range for microporous materialsin an unambiguous way [101]. This approachhas been applied for zeolites [22], metal-organ-ic framework materials [105] as well is recom-mended in a very recent standard of the Inter-national Standard Organization (ISO) [106].

Also with regard to the determination of sur-face areas via the BET method it is of interestdiscuss the choice of the adsorptive. Nitrogen isusually considered the standard adsorptive, alsobecause of the availability of liquid nitrogen. Akey parameter for a proper BET analysis is theassumption of a cross-sectional area, i.e., thearea occupied by an adsorbed molecule in acomplete monolayer. It is known that the quad-rupole moment of the nitrogen molecule leadsfor instance to specific interactions with polarhydroxyl surface groups, causing an orientatingeffect on the adsorbed nitrogen molecules [106].Consequently, on polar surfaces the effectivecross-sectional area of adsorbed nitrogen issmaller than the customary value of 0.162 nm2.Indeed, recent experimental sorption studies onhighly ordered mesoporous silica materialssuch as MCM-41 suggest strongly that thecross-sectional area of nitrogen on a hydroxy-lated surface might differ from the commonlyadopted value of 0.162 nm2 [111, 112]. Based onmeasurements of the nitrogen volume adsorbedon silica spheres of known diameter.

In [108] a cross-sectional area of 0.135 nm2

was proposed. Consequently, using the stan-dard cross-sectional area (0.162 nm2) the BETsurface area of hydroxylated silica or other po-

The onset of cavita-tion depends onpore/cavity size andpore geometry forfor pore diameterssmaller than≈ 11 nm, but re-mains practicallyunchanged forsamples with largepores.

Surface area is a cru-cial parameter foroptimizing the useof porous materialsin many applica-tions.

Nitrogen is usuallyconsidered the stan-dard adsorptive,also because of theavailability of liquidnitrogen.



lar surfaces can be significantly overestimated.Hence, since argon molecule is monatomicand much less reactive than the diatomic nitro-gen molecule, argon adsorption (at 87 K) is analternative adsorptive for surface area determi-nation. Due to the absence of a quadrupolemoment and the higher boiling temperature,the cross-sectional area of argon (0.142 nm2 at87.3 K) is less sensitive to differences in thestructure of the adsorbent surface [22].

3.2 Pore Size Analysis

In absence of mesoporosity, the physisorptionisotherm is of type I with a plateau which isvirtually horizontal. In this case the adsorbedamount in the plateau region can be directlycorrelated with the micropore volume by ap-plying the so-called Gurvich rule [17 – 20].Here it is assumed that the pores are filledwith a liquid adsorptive of bulk-like properties,an assumption that does not allow for the factthat the degree of molecular packing in smallpores is dependent on both pore size and poreshape. In case of additional mesoporosity, themicropore volume can be obtained by applyingthe standard and comparison isotherm concept(e.g. t-plot), or the Dubinin-Radushkevich ap-proach [14 – 20, 114]; here it is also assumedthat the micropores are filled by a homoge-neous liquid phase with bulk-like properties.Other approaches such those of Horvath andKawazoe (HK), Saito and Foley (SF) andCheng-Yang (CY) [19, 20, 114, 115] allow to ob-tain the pore size distribution in addition tothe pore volume, but rely on similar macro-scopic, thermodynamic assumptions concern-ing the nature of confined adsorbate. Thisleads to inaccuracies in the pore volume andpore size determination. In case of mesoporos-ity, the total pore volume via Gurvich rule is de-termined from the adsorbed at relative pres-sure 0.95 in case of type isotherms with H1and H2 hysteresis loops. Pore size analysis ofmesoporous materials can be performed withmethods based on the the macroscopic Kelvinequation, e.g., Barett-Joyner-Halenda (BJH ap-proach) [19, 20, 66]). Direct experimental testsof the validity of the Kelvin equation weremade possible by using for instance MCM-41silica as a model material, which consists of anarray of independent cylindrical pores (of thesame diameter in the range 2 nm to 10 nm).Because of the high degree of order, the porediameter can be derived by independent meth-ods (based on X-ray-diffraction, high-resolutiontransmission electron microscopy etc.). It wasfound that the BJH method and related proce-dures based on a modified Kelvin equation can

underestimate the pore size by up to 20 – 30 %for pores smaller than 10 nm (for a compre-hensive review, please see [23] and referencestherein). This deviations are caused by a seriesof problems, including the fact that the assess-ment of pre-adsorpbed film thickness also be-comes problematic when the pore diameter de-creases [23, 20]. An improvement for pore sizeanalysis can be obtained by calibrating the Kel-vin equation using a series of MCM-silicas ofknown pore diameter (obtained from XRD in-terplanar spacing and the mesopore volume).In this manner, a relation between capillarycondensation pressures and pore size can beestablished and used to obtain an empiricallycorrected Kelvin equation valid over the cali-brated range (∼ 2 – 10 nm) [118, 119].

It is further evident that the Kelvin conceptfails to describe correctly the peculiarities ofthe critical region and the confinement-in-duced shifts of the phase diagram (i.e., criticalpoint shifts, freezing point and triple pointshifts, etc) of the pore fluid [23]. The thermody-namic state and the thermophysical propertiesof the adsorbed pore fluids, as already indi-cated, differ significantly from the bulk fluid,and this has a pronounced effect on the shapeof the adsorption isotherm; e.g., the disappear-ance of hysteresis with decreasing pore size (atgiven temperature), or increasing temperature(for a given pore size) cannot be described bythe Kelvin equation [23]. However, microscopicmethods based on statistical mechanics whichcan describe the configuration of the adsorbedphase on a molecular level (DFT, molecularsimulation) take this into account.

It has been shown that the non-local densityfunctional theory (NLDFT) with suitably cho-sen parameters of fluid-fluid and fluid-solid in-teractions quantitatively predicts the positionsof capillary condensation and evaporation tran-sitions of argon and nitrogen in cylindrical andspherical pores of ordered mesoporous mole-cular sieves (e.g., MCM-41, SBA-15, SBA-16,and hierarchically structured silica materials),[e.g., 31, 32, 120]. To practically apply this ap-proach for the calculation of the pore size dis-tributions from the experimental adsorptionisotherms, theoretical model isotherms have tobe calculated using methods of statistical me-chanics. In essence, these isotherms are calcu-lated by integration of the equilibrium densityprofiles of the fluid in the model pores. A setof isotherms calculated for a set of pore sizesin a given range for a given adsorbate is calleda kernel, and can be regarded as a theoreticalreference for a given adsorption system. Sucha kernel can then be used to calculate pore sizedistributions from adsorption/desorption iso-therms measured for the corresponding sys-

Argon adsorption at87 K is an alterna-tive adsorptive forsurface area deter-mination.

The Kelvin conceptfails to describecorrectly the pecu-liarities of the criti-cal region and theconfinement-in-duced shifts of thephase diagram.

A DFT kernel can beregarded as a theo-retical reference fora given adsorptionsystem and can thenbe used to calculatepore size distribu-tions from adsorp-tion/desorption iso-therms measuredfor the correspond-ing systems.



tems. It is important to realize that the numer-ical values of a given kernel depend on a num-ber of factors, such as the assumed geometri-cal pore model, values of the gas-gas and gas-solid interaction parameters, and other modelassumptions. The calculation of pore size dis-tribution is based on a solution of the IntegralAdsorption Equation (IAE), which correlatesthe kernel of theoretical adsorption/desorptionisotherms with the experimental sorption iso-therm (for details see [21, 23, 31]). Comparingthe calculated NLDFT (fitting) isotherm withthe experimental sorption isotherm allows tocheck the validity of the calculation. Pore sizeanalysis data obtained in this way for mesopor-ous molecular sieves obtained with NLDFTmethods agree very well with the results ob-tained from independent methods (e.g. basedXRD, TEM etc.).

The application of the NLDFT for the poresize analysis of highly ordered MCM-41 mate-rials is shown in Fig. 6. Fig. 6a shows nitrogenand argon isotherms at 77 K and 87 K, respec-tively. The nitrogen isotherm is fully reversible,whereas argon adsorption shows a small butgenuine hysteresis loop, indicating the differ-ences of thermodynamics between the con-fined argon and nitrogen states. However, theNLDFT pore size distribution curves calculatedfrom the nitrogen and argon desorption iso-therms by applying dedicated NLDFT methodsassuming nitrogen (77.4 K) and argon (87.3 K)sorption in cylindrical silica pores agree per-fectly. On the other hand, the BJH pore sizedistribution obtained from the reversible nitro-gen isotherm underestimates significantly thepore size.

Another major advantage of methods basedon DFT and molecular simulation is that theyallow to obtain an accurate pore size analysis

over the complete micro/mesopore size rangewith a single method as demonstrated inFig. 7, which shows argon sorption at 87.3 K ina micro-mesoporous ZSM-5 zeolite and result-ing pore size analysis. A type H4 hysteresisloop has been observed, with the characteristicstep down at rel. pressures ≈ 0.4. The pore sizedistribution (psd) was obtained by applying ahybrid NLDFT method which assumes argonadsorption in a cylindrical, siliceous zeolitepore in the micropore range, and a amorphous(cylindrical) silica pore model for the mesoporerange . Two different types of hybrid kernel foradsorption and desorption branches were ap-plied. The adsorption branch kernel takes cor-rectly into account the delay in condensationdue to metastable pore fluid, whereas a equili-brium NLDFT kernel was applied to the deso-rption branch. The NLDFT pore size distribu-tion clearly shows two distinct groups of pores:micropores of the same size as in ZSM-5(0.52 nm) and primary mesopores in the porediameter range from 2 – 4 nm.

The pore size distribution curves obtainedfrom adsorption and desorption branchesagree with exception of the PSD artifact ob-tained from the section of desorption branchwith the characteristic step-down in the de-sorption isotherm. As discussed in Sect. 2.2,this step down is not associated with the eva-poration of pore liquid from a specific group ofpores, i.e., the spike in the desorption pore sizedistribution curve (PSD) reflects an artifact,caused by the spontaneous evaporation of me-tastable pore liquid (cavitation, i.e., the tensilestrength effect). In contrast, the PSD derivedfrom the adsorption branch does not revealthis artifical PSD peak.

While NLDFT has been demonstrated to bea reliable method for the characterization of a

0 0.2 0.4 0.6 0.8 1Relative Pressure P/P0

0

100

200

300

400

500

600

700

800

900

Vol

ume

[cm

3 g-1

] STP

N2 (77.4 K) - AdsN2 (77.4 K) - DesAr (87.3 K) - AdsAr (87.3 K) - Des

a) b)

Figure 6. a) Nitrogen and argon sorption isotherms at 77 K, and 87.3 K, respectively in MCM-41 silica and NLDFT fitb) NLDFT (from argon and nitrogen isotherms) and BJH (from nitrogen isotherm) pore size distribution curves.

Methods based onDFTand molecularsimulation allow toobtain an accuratepore size analysisover the completemicro/mesoporesize range with asingle method.



variety of ordered and hierarchically structuredmaterials, a drawback of the standard NLDFTis that they do not take sufficiently into ac-count the chemical and geometrical heteroge-neity of the pore walls (e.g., of carbon material-s).Usually a structureless (i.e., chemically andgeometrically smooth) pore wall model is as-sumed. The consequence of this mismatch be-tween the theoretical assumption of a smoothand homogeneous surface and the experimen-tal situation is that the theoretical DFT adsorp-tion isotherms exhibit multiple steps asso-ciated with layering transitions related to theformation of a monolayer, second adsorbedlayer, and so on. Experimentally, stepwise ad-sorption isotherms are observed only at lowtemperatures for fluids adsorbed onto molecu-larly smooth surfaces, such as mica or gra-phite. However, in amorphous porous materi-als layering transitions are hindered due toinherent energetic and geometrical heteroge-neities of real surfaces. The layering steps onthe theoretical isotherms can cause artificialgaps on the calculated pore size distributions,because the computational scheme, which fitsthe experimental isotherm as a linear combi-nation of the theoretical isotherms in indivi-dual pores, attribute a layering step to a porefilling step in a pore of a certain size. The prob-lem is enhanced in many porous carbon mate-rials, which exhibit in contrast to micro-meso-porous zeolites broad PSD’s, and here theartificial layering steps obtained in the theore-tical isotherms cause artificial gaps on the cal-culated pore size distributions. This problemhas been addressed by various approaches[121 – 124] including the so-called QSDFT(quenched solid density functional theory)[122, 123]. QSDFT allows to take into accountwall heterogeneity in a straightforward way. Ithas been shown that QSDFT significantly im-proves the method of adsorption porosimetryfor heterogenous porous carbons, the pore sizedistribution (PSD) functions do not possessanymore the artificial gaps in the regions of∼1 nm and ∼2 nm [123]. This is demonstratedin Fig. 8 in which the pore size distribution cal-culations for active carbon fiber ACF-15 for thenitrogen adsorption isotherm are presentedand the QSFDFT and NLDFT results are com-pared. Fig. 8a shows the fit of the experimentalisotherm with the calculated theoretical one.The QSDFT provides a significant improve-ment in the agreement between the experi-mental and the theoretical isotherms, in parti-cular in the low pressure range of themicropore filling (Fig. 8a). The prominent stepat P/P0 ∼ 3·10–4 that is characteristic to the the-oretical NLDFT isotherms, is due to the mono-layer transition on the smooth graphite sur-

face, is completely eliminated in the QSDFTisotherm. As a consequence, a sharp mini-mum in the NLDFT pore size distributioncurve at ∼1 nm, which is typical to the NLDFTpore size distribution curves for many micro-porous carbons, does not appear in QSDFTcalculations (Fig. 8b). This confirms that thisminimum on the differential NLDFT pore sizedistribution is indeed an artifact caused by themonolayer step in NLDFT approach, which oc-curs at the same pressure as the pore filling ina ∼1 nm slit pore. The pore size distributioncurves in Fig. 8b show that QSDFT and NLDFTagree beyond the regions where artificial gapswere observed with NLDFT. It clearly followsthat the application of QSDFT leads to majorimprovements in the pore size analysis of na-noporous carbon materials.

However, it should be noted that the applica-tion of QSDFT and other of these advanced

Figure 7. a) Nitrogen adsorption in micro-mesoporous ZSM5 zeolite.b) NLDFT pore size analysis by applying the NLDFT metastable adsorption branchkernel on the adsorption isotherm and the equilibrium transition kernel on the deso-rption data.

a)

b)

QSDFTsignificantlyimproves the meth-od of adsorptionporosimetry for het-erogenous porouscarbons.



methods is useful and leads to accurate resultsonly, if the given experimental adsorptive/ad-sorbent system is compatible with the chosenNLDFT or GCMC kernel.

5 Summary and Conclusion

The major progress made in the understandingof the adsorption, pore condensation and hyster-esis behavior of fluids in nanoporous materialshas led to major advances in the structural char-acterization by physical adsorption, also becauseof the development and availability of advancedtheoretical procedures based on statistical me-chanics (e.g., Non-Local Density FunctionalTheory (NLDFT)) and molecular simulation.Contrary to classical, macroscopic thermody-namic approaches, these microscopic methodsdescribe the configuration of the adsorbed

phase on a molecular level. The validity of theseadvanced models (in particular NLDFT) for poresize analysis could be confirmed with the helpof ordered mesoporous molecular sieves ofknown pore size and structure. It has been de-monstrated that the application of these noveltheoretical and molecular simulation basedmethods leads to: (i) a much more accurate poresize analysis, and (ii) allows performing poresize analysis over the complete micro/mesoporesize range. NLDFT is meanwhile widely usedfor pore size analysis, featured in an ISO stan-dard and commercially available. While NLDFThas been demonstrated to be a reliable methodfor characterization of ordered and hierarchi-cally structured materials, a drawback of thestandard NLDFT is that they do not take suffi-ciently into account the chemical and geometri-cal heterogeneity of the pore walls. These defi-ciencies are currently being addressed byvarious scientific groups; a novel DFT method,namely QSDFT (quenched solid density func-tional theory ) accounts for the surface geometri-cal in-homogeneity in form of a roughness para-meter. Application of QSDFT leads to majorimprovements in the pore size analysis of nano-porous carbon materials.

More recently, the focus has shifted towardsthe structural analysis of advanced micro-me-soporous materials (e.g., micro-mesoporouszeolites, and hierarchically structured porousmaterials), which have many potential applica-tions (e.g., in catalysis, separations, etc.). Acombination of various phenomena includingmicropore filling, pore condensation, poreblocking/percolation and cavitation inducedevaporation can be observed, which is reflectedin characteristic types of adsorption hysteresis.These complex hysteresis loops introduce ofcourse a considerable complication for poresize analysis, but if interpreted correctly, alsoallow to obtain important and unique informa-tion about the pore structure of such advancedmicro-mesoporous material.

In addition to nitrogen adsorption at 77 K, itis suggested to use complimentary probe mo-lecules (e.g., argon at 87 K) not only to checkfor consistency, but also to obtain more accu-rate and comprehensive surface area, pore sizeand pore structure information. Within thisthe importance of coupling gas adsorptionwith other experimental techniques (e.g. x-rayand neutron scattering based techniques) forstudying details of the adsorption and phasebehavior of fluids in complex pore networksneeds to be pointed out.

Despite the progress made in theoretical andmolecular simulation based approaches to de-velop more realistic adsorbent models, thereare still major problems in the characterization

a)

b)

Figure 8. Comparison of the QSDFT and NLDFT methods for nitrogen adsorption foractivated carbon fiber ACF-15.a) Experimental isotherm (in semi-logarithmic scale) together with the NLDFT andQSDFT theoretical isotherms.b) NLDFT and QSDFT differential pore size distributions.



of disordered porous materials and mesopor-ous with imhomogeneous surface chemistry(incl. materials with chemically functionalizedsurfaces,). New challenges are also associatedwith just emerging new types of porous mate-rials, such as metal-organic framework materi-als covalent organic frameworks (COFs), aswell as materials with non-rigid pore struc-tures. This needs to be addressed in future ex-perimental and theoretical work with advancedtheoretical, computational and experimentalapproaches, and well chosen model materials.

I would like to thank Maritza Roman forhelp with the graphics.

Dr. M. Thommes([email protected]),Quantachrome Instruments1900 Corporate DriveBoynton BeachFl-33426, USA.

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Matthias Thommes received his Doctoratein Physical Chemistry with Prof. G.H. Fin-denegg at Ruhr University Bochum (1989 –1991) and Technical University Berlin (1992/93). From 1992 – 1995 Dr. Thommes was aResearch Associate at the Iwan-N.-StranskiInstitute of Physical and Theoretical Chem-istry, Technical University Berlin and ProjectScientist for a microgravity experiment oncritical adsorption. From 1996 to 1997, hewas an ESA fellow (European Space Agency)and Postdoctoral Research Associate at the

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Physical Adsorption Characterization of Nanoporous · PDF filePhysical Adsorption Characterization of Nanoporous Materials Matthias Thommes During recent years, major progress has

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