Deformation of Porous Carbons upon Adsorption - ACS Publications

Feb 22, 2011 - Micromeritics Instrument Corp., Inc., Norcross, Georgia 30093, United States. bS Supporting Information. ABSTRACT: N2 and CO2 sorption ...
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Deformation of Porous Carbons upon Adsorption Christian Balzer,† Timo Wildhage,† Stephan Braxmeier,† Gudrun Reichenauer,*,† and James P. Olivier‡ † ‡

Bavarian Center for Applied Energy Research, Am Hubland, 97074 W€urzburg, Germany Micromeritics Instrument Corp., Inc., Norcross, Georgia 30093, United States

bS Supporting Information ABSTRACT: 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.

1. INTRODUCTION Recording adsorption isotherms is a well-established method for the characterization of porous solids. Up to now, the solid backbone of the material under investigation is assumed to be rigid for the evaluation of the isotherms. In 1927 it was the first time that deformation of a porous solid as a result of an adsorption effect was reported.1 Later, in part, very systematic studies were performed to investigate the deformation of porous amorphous carbons upon adsorption.1-9 In 1986 Kawaguchi et al.10 monitored the deformation of a silica gel upon desorption of the pore liquid. On the basis of the findings of Scherer et al. in the 1990s, Reichenauer et al.11-13 investigated the deformation of mesoporous silicas as a result of capillary condensation with a dilatometric setup integrated into the sample holder of a commercial volumetric sorption instrument. In parallel, Baklanov et al.14 applied ellipsometry on porous layers detecting similar effects during filling of mesopores. In 1997, Tvardovski et al. reported on both swelling and contraction observed upon an in situ length change experiment with different adsorbates in porous carbon.15 Similar effects were also observed for zeolites.16 While the first theory by Bangham and Fakhoury17 to describe the deformation phenomenon in carbons at low relative pressures (below the range of mesopore filling) was based on the adsorption on a plain surface, recent theoretical approaches revealed the strong correlation between adsorption in micropores18 and the macroscopic deformation of the corresponding materials.19-21 Ravikovitch and Neimark presented nonlocal density functional r 2011 American Chemical Society

theory (DFT) calculations that reproduce adsorption and strain isotherms observed experimentally for zeolite X.20 Kowalczyk et al.22 in particular showed that deformation effect caused by adsorption in micropores may be highly sensitive to the micropore size distribution of the material under investigation and the kind of adsorbate applied. Both theoretical approaches were able to reproduce the often experimentally observed transition from contraction to expansion of carbons in the low relative pressure range. While the deformation effects observed for compliant mesoporous materials amount up to 50% relative volume change upon capillary condensation,11 the magnitude of the adsorptioninduced macroscopic strain due to micropore filling is usually on the order of 10-3 to 10-4.23 Although these are very small changes only, the local deformation can largely affect the adsorption kinetics, e.g., in terms of a change in pore openings, resulting in highly selective gate effects. In addition, the sensitivity of the deformation on the pore size provides a complementary approach to probe micropore size distributions.19,20 Recently the topic of adsorption-induced deformation attracted much attention from the geological point of view,24-37 since this behavior is relevant for the process of CO2 sequestration and methane recovery. The goal of our work was to implement and test a setup for the in situ measurement of relative length changes upon adsorption Received: November 9, 2010 Revised: January 14, 2011 Published: February 22, 2011 2553

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down to 77 K in a commercial volumetric sorption instrument; hereby a resolution in relative length change of 10-4 should be achieved. With this setup we performed systematic measurements on sol-gel-derived, monolithic porous carbons (carbon xerogels) as model systems in order to check which model(s) for adsorption-induced deformation correctly describes the observed relative length changes.

2. THEORETICAL BACKGROUND 2.1. External Surface Area - Energetic Considerations. The presence of an interface is always associated to an interfacial energy. With specific energies on the order of several joules per square meter, nanoporous solids with large specific surface areas contain high specific interfacial energies. To energetically minimize their state materials try to reduce the interface in favor of a compression of their solid phase. This means that porous solids with no adsorbate present (degassed) are in a comparatively highly prestressed state. The first theory for adsorption induced length change was proposed by Bangham and Fakhoury in 1931.17 They assumed the change of surface energy ΔF of the adsorbent in the process of adsorption to be directly proportional to the observed relative length change ΔL/L0:

ΔL ¼ λ 3 ΔF L0

ð1Þ

where λ is a constant. According to Boyd and Livingston, ΔF can be calculated from the experimental sorption data:38 Z p R3T Vads ðp0 Þ 0 ΔF ¼ jFðpÞ - F0 j ¼ dp ð2Þ p0 SBET 3 Vmol;gas 0 The change of surface energy ΔF depends on the mass specific volume of gas adsorbed Vads, the gas pressure p, the molar volume of the gas Vmol,gas, the temperature of the measurement T, and the mass specific Brunauer-Emmett-Teller (BET) surface area SBET; R is the ideal gas constant. In 1986 Scherer deduced a relationship for the dependence of the constant λ (eq 1) on the material properties of a porous adsorbent:39 λ¼

SBET 3 Fsc f Esc 3

ð3Þ

Here Fsc is the density, Esc is the Young’s modulus of the (microporous) skeleton of the carbon, and f takes into account the microscopic model structure (aspect ratio of backbone structure) and the Poisson’s ratio of the adsorbent. 2.2. Micropore Effect. Recent theoretical studies of the deformation effect occurring in the progress of micropore filling consider the adsorption process on the atomic scale with molecular simulations19,22 and DFT20,40 assuming the total free energy resulting from the sum of the elastic and the adsorption free energy. The theoretical studies reveal that a systematic nonmonotonic change in the so-called solvation or disjoining pressure σsol,40 in particular the fact that it can take positive or negative values depending on the size of the micropores present in the system under investigation, explains why both swelling as well as contraction effects may be observed experimentally upon micropore filling. Recently, Kowalzcyk et al. also analyzed the impact of adsorbate orientation on the solvation pressure.22

The microscopic stress translates into a macroscopic length change ΔL/L0 that can be approximated for small changes by ΔL 1 ΔV 1 1  3  3 σ sol ð4Þ L0 3 V0 3 K Here ΔV/V0 is the relative change of the volume of the porous sample, and K is its effective macroscopic modulus of compression. 2.3. Mesopore Effect. According to Reichenauer et al. the capillary pressure σcap, causing mesoporous materials to contract during capillary condensation and to re-expand when the mesopores are filled, is given by11 σcap ðp=p0 Þ ¼

R 3 T 3 φmeso jðp=p0 Þ 3 lnðp=p0 Þ Vmol;ads 3

ð5Þ

σcap depends on the relative pressure p/p0, the mesoporosity φmeso, the molar volume Vmol,ads of the adsorbate in liquid state, and the fraction of mesopore filling j as a function of relative pressure. The fraction of mesopores filled can be calculated according to jðp=p0 Þ ¼

Vads ðp=p0 Þ - Vh Vads ðp=p0 ¼ 1Þ - Vh

ð6Þ

where Vh is the specific volume of adsorbed gas at the low pressure closure of the sorption hysteresis. In analogy to eq 4, the relative volume change ΔV/V0 as a result of the capillary pressure σcap can be expressed by   σcap ΔV ¼ exp -1 ð7Þ V0 K For relative length changes ΔL/L0 , 1, the deformation can be written as R T φ ΔL ðp=p0 Þ  3 3 meso 3 jðp=p0 Þ 3 lnðp=p0 Þ L0 3K 3 Vmol;ads

ð8Þ

A recent publication by Gor and Neimark41 provides the first steps toward an unification of the deformation caused by capillary condensation and the change of surface energy.

3. EXPERIMENTAL SECTION 3.1. Set-Up. The base for the investigation of the deformation of porous carbons upon gas ad- and desorption was a commercial volumetric sorption instrument (Accelerated Surface Area and Porosimetry Analyzer (ASAP 2020) by Micromeritics, Norcross, GA, USA). Within the framework of this study, the sorption measurements were performed with N2 (purity 5.0) at 77 K and CO2 (purity 4.5) at 273 K, respectively. To enable in situ dilatometry, the original sample holder of the instrument was replaced by a customized holder (see Figure 1S, Supporting Information), which allows mounting of the sensor unit of a linear variable differential transformer (LVDT by Macro Sensors) on the outside of the holder. The core of the LVDT setup consisting of a short metal rod is located inside the customized sample holder and is connected to the top of the monolithic sample by means of a glass rod (see Figure 1S). To suppress effects due to temperature changes, the LVDT was surrounded by a temperature-controlled shield. A similar dilatometric measurement device was first used in 1994 by Tvardovski et al.42 This setup allows measuring of length changes with an accuracy of 0.2 μm. As the sample length L0 can be up to 100 mm, the resolution in relative length change can be as high as 2  10-6. The minimal sample length depends on the resolution in relative length change required; with about 1 mm, the resolution in relative length change achieved with the current setup is only about 2  10-4. 2554

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Figure 1. SEM images of the three samples investigated taken with the same magnification. The inset in the far right image shows a close-up of the mesostructure of sample C5050(HT). After the completion of each sorption run, the recorded length change data are filtered to identify the values corresponding to the equilibrated state within each dosing step; the equilibrium data are then correlated with the corresponding sorption isotherm by a software routine, and a length change isotherm is created. The samples used were monolithic carbon carbons with a diameter of 3 mm and a length between 33 and 73 mm. Prior to every measurement the samples were degassed in the sample holder for at least 3 h at about 600 K under vacuum. To ensure a constant temperature during measurement, the sample holder was thermostatted. In the case of N2, the cryogen was liquid nitrogen with a temperature of 77 K; the CO2 measurements were performed at 273 K. Since the gas pressure within the ASAP 2020 instrument is limited to about 1200 mbar, the temperature of the cryogen and the gas type determined the experimentally accessible relative pressure range. 3.2. Model Materials Applied. The effects of deformation upon adsorption were studied using monolithic rods of carbon xerogels as model systems. Carbon xerogels consist of a three-dimensionally interconnected network formed by microporous particles arranged in chains. These amorphous synthetic porous carbons were derived by pyrolysis of organic xerogel precursors. The precursors were synthesized via a solgel process, which is described in detail elsewhere (e.g., ref 43). Briefly, an aqueous solution of resorcinol and formaldehyde was prepared; hereby the dilution of the reactants resorcinol and formaldehyde in the starting aqueous solution was adjusted to a mass ratio M = 30 or 50 (M = mass of resorcinol and formaldehyde to total mass of the solution). Then sodium carbonate was added as a catalyst. The amount of catalyst controls the particle growth during the chemical reaction and thus the size and specific external surface area of the backbone forming particles; at a given porosity of the xerogel, the catalyst concentration therefore also determines the average pore size. The molar ratio of resorcinol to catalyst was set to 5000. The solution was filled in glass tubes with an inner diameter of 4-5 mm and sealed. Then the samples were exposed to 85 °C for 24 h for gelling and curing. Afterward, the water in the pores of the wet gel was replaced by ethanol to reduce the surface tension upon drying. Subsequently, the gels were dried under ambient conditions. Finally the resulting organic xerogels were pyrolyzed in an argon atmosphere at 800 °C. This way three samples were synthesized with a molar ratio of resorcinol to catalyst of 5000. Two of them had a mass ratio of M = 50 and one of M = 30. According to their synthesis parameters, the samples are labeled C5030 and C5050. To investigate the impact of the micropore characteristics on the deformation, one of the C5050 samples was subjected to a thermal treatment at 1800 °C under argon (sample labeled C5050(HT)). Hightemperature treatment is known to reduce the micropore volume accessible to the probing gas and is therefore expected to change the deformation effect in the range of micropore adsorption. A detailed study of the effects of high-temperature treatment on carbon xerogels is given in ref 43. 3.3. Characterization. The carbon xerogels were characterized by scanning electron microscopy (SEM), sound velocity measurement, small-angle X-ray scattering (SAXS), as well as N2 and CO2 sorption.

Table 1. Macroscopic Density Gmacro, Young’s Modulus E of the Macroscopic Sample, and Mean Diameter DPP of the Particles Forming the Skeleton of the Xerogels Investigated Fmacro (g/cm3)

E (GPa)

DPP (nm)

C5030

0.288 ( 0.014

0.084 ( 0.005

400 ( 100

C5050

0.729 ( 0.036

6.70 ( 0.71

15 ( 5

C5050(HT)

0.757 ( 0.038

7.24 ( 0.82

15 ( 5

sample

SEM images of all samples at the same magnification are shown in Figure 1. From the different SEM images taken, the mean diameter DPP the particles forming the skeleton of the xerogels investigated was determined. The Young’s modulus E of the macroscopic sample was calculated from the macroscopic density Fmacro and the sound velocity measured for the samples using Biot’s formula.44 The values for Fmacro, E, and DPP are given in Table 1. Table 1 as well as the SEM images in Figure 1 show that the mesostructure (skeletal particles as well as interparticular pores) of the samples C5050 and C5050(HT) are identical as expected since the hightemperature treatment affects the structure on the micropore scale only. In contrast, the skeleton of the sample C5030 is much coarser, and the macroscopic density is only a third compared to the other two samples. The N2 and CO2 sorption isotherms taken at 77 and 273 K in the course of the length change measurements were analyzed applying the BET theory, the t-plot method, and the Dubinin-Radushkevich (DR) equation. The BET theory was applied in the relative pressure range 0.01 < p/p0 < 0.1 according to the recommendations given in ref 45, yielding the mass specific BET surface, SBET. The linear region of the t-plot was used to determine the mass specific external surface Sext and the specific micropore volume Vtmp accessible to N2. The DR equation was applied to evaluate the CO2 data, thus providing the specific DR and the characteristic energy E0. The results micropore volume Vmp are summarized in Table 2. In addition, the N2 and CO2 sorption isotherms were analyzed with a commercial software (Micromeritics) to determine the pore size distribution (PSD) via DFT46 (see also section 5.2). Complementary SAXS measurements were performed at the German synchrotron lab HASYLAB, Hamburg, at the SAXS beamline B1 (JUSIFA). For the measurements, X-rays with a photon energy of 12 keV (corresponding to a wavelength of the X-rays of 1.035 Å) were used. The total range in scattering vector q covered by the experiments was 0.06-8 nm-1. Applying a glassy carbon reference, the scattering curves were calibrated on an absolute scale providing the scattering cross sections (in units of cm2 g-1 srad-1) (Figure 2). The scattering cross sections are shown in Figure 2. Using the twophase-model (e.g., ref 47), the mean density of the primary particles FPP and the total specific micropore volume VSAXS were determined (see mp Table 3). In addition, the shift of the micropore shoulder in the scattering cross section of sample C5050 (Figure 2) (at q > 1 nm-1) to lower q-values with high-temperature treatment indicates an increase in micropore size, while the shape of the cross section at q-values below 0.6 nm-1 is unaffected. 2555

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Table 2. Parameters Determined for the Three Carbons Investigated from Their Sorption Isotherm Taken for N2 at 77 K and CO2 at 273K, Respectively N2 sorption

CO2 sorption Vtmp 3

VDR mp 3

SBET

Sext

(m2/g)

(m2/g)

C5030

580 ( 20

10 ( 2

C5050

700 ( 10 200 ( 10 0.20 ( 0.01 0.26 ( 0.01 30.5 ( 0.2

sample

(cm /g)

E0

(cm /g)

(kJ/mol)

0.25 ( 0.01 0.31 ( 0.01 30.6 ( 0.2

C5050(HT) 205 ( 10 180 ( 10 0.01 ( 0.01 0.05 ( 0.01 25.5 ( 0.2

Figure 2. Differential SAXS cross sections for the three carbon xerogels investigated.

Figure 3. Nitrogen sorption isotherms taken at 77K (top) and simultaneously measured length change isotherms (bottom).

Table 3. Density Gpp and Specific Micropore Volume VSAXS mp of the Microporous Carbon Backbone as Derived from SAXS Data

determined in intervals of 5 h in the course of the measurement. The N2 sorption isotherms for all samples are shown in Figure 3. Apart from the volume adsorbed at very low relative pressures (micropore filling), the adsorption behavior of the samples C5050 and C5050(HT) is very similar, while sample C5030 shows large differences in both, the multilayer adsorption and the range of mesopore filling. This is in line with the structural differences of the xerogel skeletons that appear in the SEM images (Figure 1). The desorption branches of the isotherm were only measured for the samples C5050 and C5050(HT) since the sample C5030 is free of mesopores and therefore exhibits no capillary condensation. The macroscopic deformations of the samples as measured by in situ dilatometry are shown in Figure 3. The data provide the following trends: Samples C5050 and C5030 show clearly visible expansion effects of up to about 2 to 3  10-3 in relative length change in the scope of micropore filling (p/p0 < 10-3); hereby a small initial contraction is followed by an expansion of the carbons. The sample with the high-temperature treatment C5050(HT) reveals a more pronounced contraction, however, a maximum expansion that is about an order of magnitude smaller than for the other two samples. Expansion continues in all cases for p/p0 > 10-3 in the course of multilayer adsorption; hereby the carbons C5050 and C5030 show a very similar behavior despite the large structural differences between these two samples on the mesoscale. For the hightemperature-treated porous carbon (C5050(HT)), the observed expansion is significantly smaller than that for its reference C5050, despite similar specific external surface areas. For p/p0 > 0.7, capillary condensation is accompanied by mesopore deformation for C5050 and C5050(HT). Compared

sample

FPP (g/cm3)

3 VSAXS mp (cm /g)

C5030

1.28 ( 0.06

0.33 ( 0.04

C5050

1.34 ( 0.07

0.29 ( 0.04

C5050(HT)

1.31 ( 0.07

0.31 ( 0.04

The structural characterization reveals by roughly a factor of 20 larger backbone particles for sample C5030 compared to the samples C5050 and C5050(HT) (Table 1) and accordingly a smaller external surface area (Table 2). The specific micropore volume present is very similar for all samples (Table 3), although the high-temperature treatment strongly reduces the accessibility of the micropores for the analysis gases applied (Table 2). The large difference in the macroscopic Young’s modulus between the samples C5030 and C5050 is essentially an effect of the large difference in density.

4. RESULTS With the setup described in section 3.1, sorption measurements with in situ dilatometry were performed for all three samples. The analysis gases used were N2 and CO2. Each measurement provides simultaneously a sorption and a length change isotherm. The relative pressure range covered depends on the combination of analysis gas and temperature of the cryogen. 4.1. N2 at 77 K. The measurements with nitrogen were performed at 77 K. Under these conditions the entire relative pressure range of nitrogen between p/p0 = 10-6 and p/p0 = 1 is accessible. To ensure exact measurements, especially for relative pressures close to 1, the saturation pressure of N2 was

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Figure 5. Relative length change of the three carbons investigated as a function of the change in surface energies calculated according to eq 2.

Table 4. Parameter λ Determined by a Fit of eq 1 to the Experimental Data and Elastic Modulus Es of the Xerogel Backbone Calculated with the Structural Factor f According to eq 3 sample

Figure 4. CO2 sorption isotherms taken at 273 K (top) and simultaneously measured length change isotherms (bottom); the details of the length change isotherm in the low relative pressure regime are shown in Figure 2S.

to the length change observed upon micropore filling and multilayer adsorption, the deformation of the samples during mesopore filling is quite small. C5030 exhibits no capillary condensation and therefore no mesopore deformation. 4.2. CO2 at 273 K. The measurements with CO2 were performed at 273 K. The accessible relative pressure range is about 10-6 to 0.03. CO2 measurements therefore give a very detailed picture of the pressure range associated with micropore adsorption. The sorption and length change isotherms of the three porous carbons are compiled in Figure 4. The CO2 sorption measurements show essentially the same trend as detected for N2 as analysis gas: C5030 and C5050 adsorb very similar amounts of gas, while there is barely adsorption in micropores for the sample C5050(HT). The relative length change is also qualitatively very similar to the trends observed for N2: C5050 and C5030 both exhibit a small contraction before they expand with further increase in relative pressure. The contraction of C5050(HT) is more pronounced compared to the other samples, with the subsequent expansion being very small and merely compensating the contraction.

5. DISCUSSION 5.1. Energetic Approach. To test the energetic approach of Bangham and Fakhoury to explain the observed relative length change of the samples, the change of the surface energy ΔF was calculated according to eq 2 from the adsorption data of the N2 isotherms and plotted against the relative length change ΔL/L0 measured (see Figure 5). For surface energies ΔF > 10 mJ/m2 the data show clearly a linear behavior. Below ΔF = 10 mJ/m2

λ (m2/J)

f

C5030 C5050

0.735 ( 0.004 0.721 ( 0.003

C5050(HT)

0.724 ( 0.003 0.007 ( 0.001

ES (GPa)

E (GPa)

0.040 ( 0.002 13.6 ( 1.1 0.084 ( 0.005 0.046 ( 0.002 14.8 ( 1.1 6.70 ( 0.71 28.3 ( 4.6

7.24 ( 0.82

eq 1 does not hold, which is likely due to the fact that this range is dominated by micropore adsorption. The small deviations from the linear increase of the relative length change above ΔF = 60 mJ/m2 in the case of the samples C5050 and C5050(HT) are caused by deformation effects due to capillary condensation. The fit of the experimental data (Figure 5) yields the proportionality constant λ in eq 1 that contains structural parameters of the sample (see eq 3). With the structure factor f calculated according to ref 39, assuming a Poisson’s number of 0.2,48 and an aspect ratio determined from the porosity of the sample, the mass specific surface area SBET, and the density of the primary particles FPP, the Young’s modulus of the skeletal structure ES was determined from λ. The values for f, λ, ES, and the macroscopic Young’s modulus E are given in Table 4. Despite the big differences in their macroscopic Young’s moduli E, the skeletal moduli of the samples C5030 and C5050 are identical within the range of experimental uncertainty. This is expected since both xerogel backbones consist of microporous carbon with similar micropore characteristics (micropore volume and characteristic energy); obviously the size of the backbone particles does not have a relevant impact on their Young’s modulus. On the other hand, the skeletal modulus ES of the sample C5050(HT) is roughly doubled with the high-temperature treatment that significantly changes the size and the structural arrangement of the microcrystallites within the backbone particles.49 This observation may be quite valuable for the prediction of the thermal conductivity of carbon-xerogels treated at temperatures well above 1000 °C. Figure 3 shows that the meso- and the macroporous carbons C5050 and C5030 show almost identical relative length changes, although their external surface area differs by about a factor of 20 (Table 2). The external surface seems to be only of marginal 2557

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Figure 6. PSDs calculated from the CO2 isotherms using DFT.

Figure 7. Solvation pressure σsol for adsorption of CO2 at 333 K in carbon versus width of the slit-shaped pores. The different curves correspond to calculations performed for different values of the parameter “bulk pressure” (0.03 to 27 MPa, after ref 22). The region marked in gray corresponds to the conditions covered experimentally during the measurements presented in this study.

importance for the length change isotherm, even in the relative pressure range associated with multilayer adsorption. 5.2. Micropore Deformation Effects. At surface energies below 10 mJ/m2, the linear relationship between the relative length change and the change in surface energy does not hold. To understand the length changes observed, the micropore size distributions were calculated by DFT (Figure 6). The plot shows that DFT yields very similar micropore sizes for the samples C5030 and C5050, matching the almost identical experimental length change isotherms (Figure 4). In contrast, both the isotherm as well as the corresponding PSD show that sample C5050(HT) is nearly free of accessible micropores and, as a consequence, exhibits only very small micropore deformation effects at relative pressures above 0.01. The fact that the contraction of the sample at relative pressures below 0.01 is more pronounced than for the untreated reference C5050 appears contraintuitive at this point. In the following we compare the results of the CO2-sorption measurements with theoretical predictions by Kowalczyk et al. Figure 7 shows the results for the solvation pressure σsol calculated for the adsorption of CO2 on carbon at 333 K as a function of the pore width and the bulk pressure p applied onto

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Figure 8. PSDs calculated from the N2 isotherms using DFT.

the sample. Apart from the 60 K difference in temperature, the parameters used for the simulations and the experimental conditions are quite similar. Since solvation pressure and micropore deformation are directly proportional (eq 4), a positive value of σsol causes the micropore to swell, while a negative value results in contraction. The nonmonotonic shape of the solvation pressure shows that, not only is the total amount of micropore volume present important when it comes to deformation effects, but the distribution of the micropore size has an significant impact as well. For instance, the stronger contraction of the sample C5050(HT) compared to the reference C5050 could be caused by a reduction in pore size, thus causing swelling. This way the contraction is more pronounced even when the total micropore volume is reduced. In our experimental setup, the bulk pressure is limited to a max. pressure of 1200 mbar corresponding to the gray region in Figure 7. Within the shaded region, one can distinguish between pores smaller than 0.3 nm, contributing to expansion only, and pores larger than 0.3 nm, giving rise to contraction only. These findings result in a general contradiction between the model used by Kowalczyk et al.19,22 and the model that the DFT-software is based on,46 as DFT yields pores wider than 0.38 nm only while the dilatometric measurements clearly reveal swelling effects. Kowalczyk considers the CO2 molecule to be linear and the micropores to be deformable so that even micropores smaller than 0.4 nm can host CO2 molecules. In contrast, the DFT model applied assumes a rigid backbone and the adsorptive molecule to be spherical with an effective diameter of about 0.38 nm. This shows that there is a major inconsistency in the current models that has to be taken care of. 5.3. Mesopore Effects. The PSD of all samples calculated via DFT from the N2 sorption measurements are shown in Figure 8. As expected from the N2 sorption isotherms (Figure 3), the samples C5050 and C5050(HT) exhibit very similar PSDs between 2 and 100 nm, while the sample C5030 is almost free of mesopores. Consequently, deformation caused by capillary condensation is only observed for the samples C5050 and C5050(HT). In the relative pressure range p/p0 > 0.7, where capillary condensation takes place in the carbons investigated, the length change isotherm is a superposition of the surface effect described in section 5.1 and the mesopore deformation (see also length change isotherms of C5050 and C5030 in Figure 3). To test the theoretical model proposed by Reichenauer et al. (eq 8), the deformation due to mesopore filling/emptying (ΔL/L0)meso only has to be extracted from the length change isotherm. Using the proportional factor λ from Table 4, a theoretical length change 2558

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Figure 9. Contribution to the overall length change isotherm (ΔL/L0) (Figure 3) upon N2 sorption at 77 K that corresponds to mesopore filling only: (ΔL/L0)meso = (ΔL/L0) - (ΔL/L0)theo, with (ΔL/L0)theo accounting for the contribution due to micropore and multilayer sorption effects. Calculations are based on desorption data only. The offset of the curves is due to micropore-induced deformation effects.

isotherm (ΔL/L0)theo accounted for micropore and multilayer sorption effects was calculated. (ΔL/L0)theo describes the deformation for the sample without mesopores, and the difference (ΔL/L0) - (ΔL/L0)theo should hence provide (ΔL/L0)meso. The results for (ΔL/L0)meso for the samples C5050 and C5050(HT) are shown in Figure 9. For the evaluation, the desorption branch of the length change isotherm was chosen since the mesopore deformation is more pronounced upon desorption. The shape of (ΔL/L0)meso is in accordance with the results for amorphous and highly ordered mesoporous silica.11-13,50 Both samples exhibit a relative contraction of about 10-4 followed by reexpansion. The relative pressure of about 0.88 at maximum contraction corresponds to the point where the plateau of the isotherm at high relative pressures crosses over to the steep desorption branch of the hysteresis (Figure 3). The maximum deformation is about a factor of 1.5 smaller for the high-temperature-treated sample compared to the reference C5050. At almost identical mesopore size, the difference in deformation indicates a higher stiffness of the microporous backbone of C5050(HT). To quantitatively test the evaluation according to eq 8, the mesopore deformation (ΔL/L0)meso is plotted versus jmeso(p/p0) 3 ln(p/p0). The value for Vh required to calculate jmeso(p/p0) (eq 6) was estimated from the sorption isotherm (Vh = 240 cm3(STD)/g for the sample C5050 and Vh = 120 cm3(STD)/g for the sample C5050(HT)). For the sample C5050(HT), the result is in good agreement with expectations: All points of the desorption branch down to the closure point of the hysteresis fall on one straight line. For the sample C5050, contraction and re-expansion upon desorption yield two separate straight lines. The lower of these two lines corresponds to the plateau of the hysteresis loop (p/p0 > 0.88) and is approximately independent of Vh while the upper line (corresponding to the steep branch of the hysteresis loop, 0.7 < p/p0 < 0.88) is highly sensitive to the value of Vh. To force both lines to coincide, an unphysical value of Vh (350 cm3(STD)/g) is necessary. The reason for this discrepancy is not yet understood. From the proportionality factors in Figure 10 the macroscopic bulk moduli K are determined (see eq 8), which in turn are used to calculate the macroscopic Young’s modulus Emeso: Emeso ¼

K ð1 - 2 3 υÞ

ð9Þ

Figure 10. Deformation due to mesopore filling (ΔL/L0)meso plotted versus jmeso(p/p0) 3 ln(p/p0) (eq 8) for the two samples with almost identical mesopore properties; in contrast to sample C5050, sample C5050(HT) was treated at 1800 °C. Calculations are based on desorption data only.

Table 5. Elastic Moduli Emeso Determined from the Relative Length Change upon Emptying of Mesopores and Sound Velocity Measurements rel. length change sample

K (GPa)

Emeso (GPa)

sound velocity E (GPa)

C5050 hysteresis branch

2.9 ( 0.2

5.2 ( 0.4

6.70 ( 0.71

plateau

4.9 ( 0.8

8.8 ( 1.5

6.70 ( 0.71

C5050(HT)

6.0 ( 0.6

10.7 ( 1.1

7.24 ( 0.82

Here, ν is Poisson's ratio. In Table 5, the results for K and Emeso (with ν = 0.20)48 are compared to the Young’s moduli E determined via sound velocity measurement. For sample C5050, both lines in Figure 10 were evaluated. The results show a qualitative agreement between elastic parameters derived from sound velocity measurements and mesopore deformation, although the values calculated from length change data are systematically higher than the ones from sound velocity measurements.

6. CONCLUSIONS Simultaneous measurements of length change and gas sorption are feasible down to 77 K by using a customized sample holder in combination with a commercial volumetric sorption instrument. Absolute length changes can be detected down to 0.2 μm. The measurements performed on different monolithic synthetic carbons used as a model system show that a length change isotherm can be considered as a superposition of several effects, each of them containing information about the material structure on a different scale. The evaluation of the swelling of the porous carbons in the range of adsorption at the external surface area yields a good correlation with the change of surface energy (Bangham’s model); the slope of these data provide a modulus of compression for the microporous backbone entities of the carbons investigated of about 20 GPa. The filling of micropores investigated with N2 and CO2 shows an initial contraction of the sample that can easily be missed if the sample is not sufficiently equilibrated. The micropore deformation observed is similar to results reported for other 2559

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Langmuir measurements on synthetic SiC-based carbons.15,23 A comparison between measured length change isotherms, DFT PSDs, and computer simulations, however, reveals discrepancies between the underlying models for micropore adsorption. The data presented show that in situ length change measurements can be considered a powerful and sensitive tool for complementary characterization of porous materials that simultaneously provide information on the elastic properties on different length scales.

’ ASSOCIATED CONTENT

bS

Supporting Information. Additional figures. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Phone: þ49-931-70564-28. Fax: þ49-931-70564-60. E-mail: [email protected].

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