Nitrogen Distribution at 77.7 K in Mesoporous Gelsil 50 Generated via

Nitrogen Distribution at 77.7 K in Mesoporous Gelsil 50 Generated via Evolutionary Minimization with Statistical Descriptors Derived from Adsorption a...
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Langmuir 2007, 23, 2145-2157

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Nitrogen Distribution at 77.7 K in Mesoporous Gelsil 50 Generated via Evolutionary Minimization with Statistical Descriptors Derived from Adsorption and In Situ SANS N. Eschricht, E. Hoinkis,* and F. Ma¨dler Hahn-Meitner-Institut Berlin GmbH, Glienickerstrasse 100, D-14109 Berlin, Germany ReceiVed September 4, 2006. In Final Form: NoVember 8, 2006 Digitized periodic material models of size 1003 nm3 of mesoporous xerogel Gelsil 50 are reconstructed by use of an evolutionary minimization technique, with two-point probability S2(r) and volume-based pore size distribution Ψ(D) as a hybrid target function. S2(r) and Ψ(D) are derived from small-angle neutron scattering (SANS) and adsorption data, respectively. The nitrogen distribution in Gelsil 50 is characterized in the multilayer adsorption and capillary-condensation regimes by S2(r) and statistical parameters obtained from in situ SANS data. The fraction of liquid-free pore space φ is calculated from nitrogen adsorption data, and the distribution Ψcorr(D) of the diameter of the liquid-free pore space at certain relative pressures is derived from Ψ(D). The evolutionary algorithm is also used to generate the spatial nitrogen distribution by means of the descriptors φ, S2(r), and Ψcorr(D). The morphological parameters obtained from the reconstructs are compared to the respective SANS results.

1. Introduction The spatial distribution of capillary-condensed liquids in porous solids affects gas transport and is therefore of interest for industrial operations such as gas purification and heterogeneous catalysis as well as for a basic understanding of confined fluids. Gas adsorption, capillary condensation, and drainage have been simulated with density functional theory (DFT), molecular simulation (MS), and other methods.1,2 Three-dimensional (3D) digitized models of the microstructure are needed for the simulation of fluid behavior with DFT and MS. The generation of atomistic 3D models by mimicking the production process is applicable to a few materials only, and the computational cost is prohibitive for models larger than ∼203 nm3. Atomistic models of controlled pore glasses with a pore size 0 we have

V ˜ (D ˜ j) )

πH ˜ (D ˜ j)D ˜ 2j 4

(9)

Taking into account Di ) D ˜ j + 2t, from eqs 8 and 9 we obtain

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Figure 1. Pore size distribution φΨ(D) for evacuated Gelsil 50 (broken line) and distributions φΨcond(D) of the diameter of the liquid-free pore space for Gelsil 50 exposed to the relative pressures P/Ps listed in the inset (continuous lines). The area under the curves represents the liquid-free pore space φ that decreases with increasing P/Ps as a result of adsorbate film formation and capillary condensation.

V ˜ (D ˜ j) V(D ˜ j + 2t)

)

˜ 2j H ˜ (D ˜ j)D H(D ˜ j + 2t)(D ˜ j + 2t)2

Figure 2. Intersection of cylindrical pores (schematic). The distribution of the diameter of the liquid-free pore space after nitrogen uptake was derived from the BJH pore size distribution for the evacuated pore system. However, the BJH model is valid for nonintersecting cylindrical pores, and the nitrogen uptake at a given P/Ps is allocated to the adsorbate film with thickness t on the interface area S of pores with a diameter D > Dcond and to the capillary condensate in pores with D e Dcond. For intersecting pores (hatched), S is lower than for nonintersecting pores; consequently, the cumulative pore volume assigned to the former is too low and needs correcting.

(10)

As long as D ˜ j > 0, the adsorbed film reduces the diameters but keeps the heights: H ˜ (D ˜ j) ) H(D ˜ j + 2t) ) H(Di). Hence, substituting eqs 6, and 7 into eq 10 yields

Ψ ˜ (D ˜ j) )

Vpore D ˜ 2j Ψ(D ˜ j + 2t) Vlf (D ˜ j + 2t)2 j ) 1,..., jmax ) imax - i0 (11)

Because the discretization length δ is cancelled out in the quotient, eq 11 also represents a continuous distribution Ψ ˜ (D), 0 < D e Dmax (substitute D for D ˜ j). The ratio Vpore/Vlf can be obtained from experimental adsorption isotherm data and is equal to 1 for the empty sample. Taking into account capillary condensation, for the distribution of the diameter D of the liquid-free space of cylindrical pores we have

{

D + 2t e Dcond 0 Ψ ˜ cond(D) ) Ψ ˜ (D) otherwise

(12)

Table 1 shows φ, t, and Dcond for the relative pressures P/Ps at which in situ SANS experiments have been performed. Figure 1 displays the distribution φΨ(D) for empty Gelsil 50 as calculated with the BJH method by use of the Micromeritics software. To have values of Ψ for all values of D necessary in the computations, the data were fitted by a polynomial of order 10 in the range of 1 e D (nm) e 8 and by a sum of two exponentials for D > 8. The φΨcond curves are plotted also. The fractions φ0 of liquidfree pore space were estimated by means of the relation

φ0 ) φ

∫DD

max

cond

Ψcond(D) dD

(13)

and are listed in column 6 of Table 1. For P/Ps > 0, one notes that φ0 is significantly smaller than φ. Equation 12 is valid only

Figure 3. Pore size distribution φΨ(D) for evacuated Gelsil 50 (broken line) and distributions Ψcorr(D) ) βφΨcond of the diameter of the liquid-free pore space corrected for intersection (continuous lines). φ is the fraction of liquid-free pore space calculated from adsorption isotherm data, and β is the intersection factor correcting the distribution such that the integral over Ψcorr equals φ.

for unconnected cylindrical pores. In Gelsil 50, however, the pores intersect; therefore, the model used for the derivation of eq 12 leads to an overestimation of the nitrogen quantity located in the multilayer or, equivalently, to an underestimation of the liquid-free pore volume. Figure 2 illustrates this effect. Note that the ratio (last column in Table 1)

β :)

φ φ0

(14)

is almost constant for relative pressures of P/Ps e 0.465, that is, up to the hysteresis loop of the isotherm. We assume that β

Nitrogen Distribution in Mesoporous Gelsil 50

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Q)

∫0∞I(q)q2 dq ) 2π2φ(1 - φ)(1Fb - 2Fb)2

(17)

Here, φ is the volume fraction of one phase, and 1Fb and 2Fb are the scattering-length densities of phases 1 and 2. A linear plot of the I(q) data showed an abrupt increase at q < 0.05 nm-1. This is certainly an artifact that is probably due to scattering at the interface between the sample holder and sample. Therefore, we discarded the experimental I(q) data for q < 0.05 nm-1. In addition, we found it to be essential to use smoothed data IS(q) for the calculation of γ0(r); see below. The IS(q) function was obtained by a smoothing cubic spline approximation to I(q).29 For q < 0.1 nm-1, we used IS(q) to extrapolate to q ) 0. We calculated the invariant from

Q ) F∆ +

∫abIS(q)q2 dq +

CP b

(18)

where a and b are the lower limits of the experimental q range and the q range satisfying Porod’s law, respectively, and F∆ is the estimate of the integral between 0 and a. Porod’s law25 is

lim I(q) ) CPq4

(19)

qf∞

Figure 4. Mean values of morphological parameters as a function of the fraction of liquid-free pore space φ. hl is the first moment of the distribution of chord length, and hllf ) hl/(1 - φ) is the mean chord length in the liquid-free volume. The scattering-length densities of silica and liquid nitrogen are about equal, so the mean chord length hlliso ) hl/φ refers to the condensed phase consisting of a silica skeleton, a liquidlike nitrogen film, and capillary condensate. D h is the mean diameter of the liquid-free volume of pores with an assumed cylindrical shape calculated from φΨ(D) and Ψcorr(D). S and SBET are the interface areas calculated from SANS data and adsorption data, respectively. The lower closure point of the hysteresis loop of the isotherm is at P/Ps ) 0.465. Data at P/Ps ) 0.618 refer to the adsorption branch of the hysteresis loop.14,19

accounts for the intersection of pores, hence we adapt Ψcond by multiplication with the “intersection factor’’ β. The corrected curves

Ψcorr(D) :) βφΨcond(D)

(15)

are displayed in Figure 3. The mean values D h of the remaining diameters are shown in Figure 4. We have used the distributions Ψcorr as one of the target descriptors for the reconstructions of the spatial nitrogen distributions; see section 5.4.

4. Morphological Parameters and Descriptors from In Situ SANS Data The intensity I(q) scattered by a 3D statistically isotropic twophase heterogeneous medium without long-range order is given by eq 16:27,28

I(q) )

∫0∞r2γ0(r)

2Q π

sin(qr) dr qr

(16)

γ0(r) is the 3D-averaged autocorrelation function of the scatteringlength density fluctuation, and r is a distance. The invariant Q is defined by25 (27) Debye, P.; Bueche, M. J. Appl. Phys. 1949, 20, 518. (28) Guinier, A.; Fournet, G. Small Angle Scattering; Wiley and Sons: New York, 1955.

The effect of the low-q data on Q is weak because F∆ is only on the order of 1% of the integral in eq 18 for monolithic Gelsil 50. Morphological parameters were calculated from invariant and Porod constants: the mean value of the chord distribution is hl ) 4Q/πCP; the mean value of the chords is hllf ) hl/(1 - φ) within the liquid-free volume (in empty pores and in pores with an adsorbate film); the mean value of the chords is hlliso ) hl/φ within the condensed phases (in the skeleton and in liquidlike nitrogen); and the interface area is S ) 4φ(1 - φ)/Fbulkhl.30 The correlation function can be calculated from the transformation

γ0(r) ) Q-1

∫0∞q2IS(q)

sin(qr) dq qr

(20)

which may be expressed by the use of Porod’s law by

γ0(r) ) Q-1

∫0bq2IS(q)

sin(qr) dq + CP qr

∫bq

q-2

max

sin(qr) dq qr (21)

The first derivative of the correlation function at r f 0 is25,31

1 hl

(22)

∫0∞γ0(r) dr

(23)

γ′0(r f 0) ) The correlation length is30

lc ) 2

We tested the capacity for the practical use of eq 21 with theoretical I(q) data for a Debye random two-phase medium (DR)27 and a Teubner-Strey two-phase medium (TS)32 with correlated disorder. For DR and TS, analytical expressions are known for I(q), γ0(r), and the first and second derivatives of γ0(r). I(q) for the TS was similar to I(q) for the original Gelsil 50. We calculated IS(q) as described above from 90 theoretical I(q) data points for the TS. We determined b and calculated γ0(r) with eq 21 for a (29) IMSL C/Math/Library; Visual Numerics Inc.: 1999; p 206. (30) Porod, G. In Small Angle X-ray Scattering; Glatter, O., Krattky, O., Eds.; Academic Press: London, 1982; p 17. (31) Schmidt, P. W. In Modern Aspects of Small-Angle Scattering; Brumberger, H., Ed., Kluwer Academic Publishers: Dordrecht, The Netherlands,1995; p 1. (32) Teubner, M.; Strey, R. J. Chem. Phys. 1987, 87, 3195.

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given r value for some qmax until γ0(r) became independent of qmax. We found qmax ) 5 nm-1 for r ) 1 nm. γ0(r) obtained with eq 21 for DR and TS agreed with the theoretical data. The twopoint probability for a two-phase system is related to φ and γ0(r) by5

S2(r) ) φ(1 - φ) γ0(r) + φ2

0 e r e rmax

(24)

The CLD A(r) describes the distribution of distances along straight lines between two smooth boundaries in a two-phase system.33-35 In contrast to p(r), the function A(r) includes all kinds of chords between interfaces (e.g., chords within the pores, within the solid, and within the pores and solid). A(r) is obtained from the mean chord length and γ′′0(r):30

A(r) ) hl γ′′0(r)

(25)

The value of A(r) at small r reflects the geometrical arrangement of interfaces. Angular regions originate from sharp edges or contact points of, for example, spherical particles. For angular regions, A(r f 0) * 0.34-36 γ′′0(r) can be calculated with an expression derived by Gille37 for scattering data that satisfy Porod’s law:

γ′′0(r) ≈

1 rQ 2

∫0∞[q4I(q)]′′

[

]

sin(qr) dq qr rmin e r(nm) e rmax (26)

We tested the capacity for the practical use of eq 26 with theoretical IS(q) for DR and TS. Analogous to eq 21, we calculated the value of the integral in eq 26 by using eq 19. One notes that the second of the integrals obtained this way from eq 26 with the limits b and ∞ is zero. Therefore, the usable q range is restricted to data for which eq 19 is not satisfied; correspondingly, eq 26 is valid for r g rmin only. Model calculations with IS(q) for TS revealed that rmin ≈ 1 nm. For r > rmin, A(r) calculated with eqs 25 and 26 and A(r) obtained by differentiating the analytical expression for γ0(r) agreed. We also calculated functions A(r) from experimental IS(q) data measured for Gelsil 50 at different P/Ps values. Recalculation of γ0(r) from γ′′0(r) and calculation of I(q) with eq 16 resulted in data that agreed poorly with the experimental scattering curves. This is due to error propagation from experimental I(q) to γ′′0(r) and error propagation in the reverse direction to I(q) (i.e., the inverse nature of the problem). Nevertheless, A(r f 0) may serve as an indicator of the change in angularity due to nitrogen adsorption in Gelsil 50. Finally, we discuss whether the equations for scattering from the two-phase systems listed above can be applied to scattering from Gelsil 50 with an adsorbed liquid-nitrogen film and some pores filled by liquid nitrogen. The scattering-length densities of the solid, adsorbate film, and capillary condensate are solFb, filmF , and conF , respectively. Complete filling with nitrogen at b b 77.7 K of all pores in CPG-10-75,16,38 Gelsil 50,39 Gelsil 75,15 and SBA-15 silica23 gives for the ratio of intensities filledI(q)/evacuatedI(q) ≈ (1 - conF /solF )2 values of ∼0.02 for Gelsil b b (33) Mering, J.; Tschoubar, D. J. Appl. Cryst. 1968, 1, 153. (34) Perret, R.; Ruland, W. J. Appl. Cryst. 1972, 5, 183. (35) Tchoubar, D. In Neutron, X-ray and Light Scattering: Introduction to an InVestigatiVe Tool for Colloidal and Polymeric Systems; Lindner, P., Zemb, Th., Eds.; North-Holland: Amsterdam, 1991; pp 157-174. (36) Smarsly, B.; Go¨ltner, C.; Antonietty, M.; Ruland, W.; Hoinkis, E. J. Phys. Chem. B 2001, 105, 831. (37) Gille, W. Eur. Phys. J. B 2000, 17, 371. (38) Hoinkis, E. In Berichte des Hahn-Meitner-Instituts Berlin; BENSC Experimental Report 1998; HMI-B559; HMI Berlin: Berlin, 1999; pp 212-213. (39) Hoinkis, E. In Berichte des Hahn-Meitner-Instituts Berlin; BENSC Experimental Report 2000; HMI-B565; HMI Berlin: Berlin, 2000; p 295.

Figure 5. Correlation functions γ0(r) calculated from in situ SANS data for Gelsil 50 exposed at 77.7 K to certain relative pressures P/Ps of nitrogen. The inset shows an expanded view of the γ0(r) curves near the minimum at rmin. The lower closure point of the hysteresis loop of the N2 isotherm is at P/Ps ) 0.465. The shift of rmin corresponds to the shift of the maximum in I(q) to lower q; see Figure 6. Capillary condensation at P/Ps > 0.465 leads to a significant change in the shape of γ0(r) and a large shift of rmin.

Figure 6. Small-angle neutron scattering intensity I(q) in arbitrary units for a cylindrical Gelsil 50 sample (d ) 5 mm, h ) 2 mm) (symbols) at the relative pressures listed in the inset. The lines represent I(q) data calculated by use of eq 16 with the correlation functions γ0(r) shown in Figure 5.

50 and ∼0.01 for CPG-10-75, Gelsil 75, and SBA-15 silica. This shows that the contrast matching condition (1 - conFb/solFb)2 , 1 is nearly satisfied for capillary-condensed nitrogen in silicas, so with respect to neutron scattering, solid plus liquid represent one phase, and the liquid-free space, the other one. It is generally

Nitrogen Distribution in Mesoporous Gelsil 50

Figure 7. Mean chord length hl and correlation length lc as a function of the film thickness t.

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Figure 9. Hybrid target function E ) ES2(r) + EΨ(D) + RF plotted as a function of the number of selection steps. It contains the (squared) distances ES2(r) ) (Sˆ 2(r) - S2(r))2 and EΨ(D) ) (Ψ ˆ (D) - Ψ(D))2, thus representing the deviation of the actual descriptors from their target values during evolutionary minimization. The “penalty term’’ RF improves the minimization process; see ref 8 for details.

Figure 10. Average of two-point probability Sˆ 2(r) obtained from 10 reconstructions of Gelsil 50 and S2(r) as calculated from correlation function γ0(r) and porosity φ. The difference between Sˆ 2(r) and S2(r) for small r is discussed in ref 8.

Figure 8. Chord-length distributions A(r) for certain relative pressures P/Ps. A(r) describes the distribution of distances along straight lines between two boundaries. For the evacuated sample, A(r) includes chords within the pores, within the solid, and within the pores and solid. The scattering-length densities of silica and liquid nitrogen are about equal (i.e., with respect to SANS, silica and liquid nitrogen represent one condensed phase). For angular regions, A(r f 0) * 0.35

assumed that the adsorbed film is liquidlike,40 but experimental evidence for this assumption is rare, except the adsorptive smoothing of the rough internal surface in CPG-10-75 observed with in situ SANS.16,41 For a rough internal surface, I(q) ≈ q-R with R < 4 for large q but R ) 4 for a smooth surface. The adsorption of contrast-matched benzene41 or nitrogen16 on CPG10-75 led to a gradual change in the scattering exponent from R ) 3.7 to 4, which indicates smoothing of the rough interface by adsorbed molecules and shows that solFb ≈ filmFb for the multilayer. Therefore, it appears to be reasonable to apply eqs (40) Cohan, L. H. J. Am. Chem. Soc. 1938, 60, 433. (41) Hoinkis, E. AdV. Colloid Interface Sci. 1998, 39, 76-77.

Figure 11. Average of geometric PSDs φΨ ˆ (D) as obtained from 10 reconstructions of Gelsil 50 and the BJH pore size distribution φΨ(D).

16-26 to scattering from Gelsil 50 with an adsorbed nitrogen film and more or less pores filled by condensed nitrogen. A review of earlier applications of the contrast-matching technique

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5. Results and Discussion

Figure 12. Number N(Z) of pores in a 1003 nm3 reconstruct of Gelsil 50 as a function of the number of adjacent pores Z. The plot averages the data of 10 reconstructs.

Figure 13. Reconstruction of Gelsil 50 and of the nitrogen distribution within the pore system. Model size, 1503 voxels; resolution, 2/3 nm per voxel. Panels a-d show a slice (of 15 nm thickness) cut parallel to one of the faces of the periodic cubic model. (a) Initial random distribution of a fraction of 0.631 voxels representing the pore space and a fraction of 1 - 0.631 voxels for the solid. (b) Empty reconstruct: white areas ) cross sections of pores; dark gray areas ) internal surface of pores; gray areas ) skeleton. Islands within some large pores represent cross sections of elevations on the pore surface. The random character of the pore system is evident; the visual appearance closely agrees with that of a TEM image of Gelsil 50.9 (c) Random distribution of nitrogen (green) within the pores at P/Ps ) 0.338, φ ) 0.370, and U ) 0.414. (d) Nitrogen distribution after reconstruction with two-point probability and size distribution of the liquid-free pore space as descriptors. Pores with diameter D e Dcond ) 2.9 nm were initially filled with nitrogen to simulate capillary condensation; however, during reconstruction, the condensed nitrogen was allowed to redistribute. Covering the pore surface with adsorbed nitrogen and capillary condensation both result in a green area in the image shown (i.e., only parts of the green areas represent capillary-condensed liquid).

to studies of the capillary condensation of water and hydrocarbons in porous solids may be found in refs 14 and 42. (42) Hoinkis, E. In Chemistry and Physics of Carbon; Thrower, P. A., Ed.; Marcel Dekker: New York, 1997; Vol. 25, pp 72-227.

5.1. Morphological Parameters. The lower and upper closure points of the triangular hysteresis loop in the nitrogen adsorption/ desorption isotherm (Figure 7 in ref 14) are at P/Ps ) 0.465 (U ) 0.529) and 0.78 (U ) 0.989), respectively. The morphological parameters vary with φ because of film formation and capillary condensation. Mean chord lengths hl, hllf, and hlliso, mean pore diameter D h , and interface areas S and SBET are plotted in Figure 4 as a function of φ. S represents the internal surface of the evacuated sample and also the interfaces of the adsorbate film and liquid menisci for P/Ps > 0. S increases with increasing φ. hlliso of the chords within liquid and solid decreases markedly with increasing φ. hllf of chords within the liquid-free pore space increases slightly with φ. The filled circle at P/Ps ) 0 represents 4Vpore/SBET ) 4.9nm. The mean chord lengths represent chords in all directions and include chords along the axis of pores. However, D h is the mean diameter of a spectrum of cylindrical pores that are used to model the real pore system, so hllf > D h within the multilayer adsorption regime at P/Ps e 0.465. With increasing pressure, an increasing number of smaller pores are filled by capillary-condensed liquid, and Ψcorr(D) becomes asymmetrical (Figure 3). D h is shifted to higher values; therefore, D h > hllf at P/Ps > 0.465. Note that hllf has been calculated from SANS data that are indeed independent of the adsorption data used to derive the BJH-PSD Ψ(D) and Ψcorr(D). The principal agreement between hllf and D h may be taken as strong support for the simple model used to derive D h . Figure 5 displays the correlation functions. The γ0(r) curves at P/Ps < 0.465 have similar shapes and form one group. The inset in Figure 5 shows a shift of the weak minimum in γ0(r) at rmin to higher r in real space, which corresponds to the shift of the maximum in I(q) at qmax to lower q in reciprocal space (Figure 6). rminqmax ≈ 2.3 for P/Ps e 0.465; that is, rmin ≈ 1/qmax. Both extrema indicate that chords of a certain length are more frequent than others; that is, there is some short-range order. We recalculated scattering curves with γ0(r) and eq 16. The upper limit of the integral was set to the distance r at which γ0(r) ≈ 0. Figure 6 displays the theoretical I(q) (lines) and the experimental I(q) (symbols). Filling of up to 53% of the available pore space with nitrogen at P/Ps ) 0.465 does not lead to a principal change in scattering behavior, as Figures 5 and 6 demonstrate, so capillary condensation in substantial volumes of the pore system does not occur, in agreement with the prediction of the Kelvin equation. We conclude that nitrogen forms a continuous film, in agreement with the accepted model of multilayer adsorption at relative pressures within the nearly linear part of the adsorption isotherm.20 A marked change in the shape of γ0(r) and in the position of rmin occurs at P/Ps ) 0.618 within the hysteresis loop (U ) 0.75); see Figure 5. At P/Ps ) 0.786 (U ) 0.989), isolated bubbles were observed by SANS.14 The effects of multilayer formation and capillary condensation on γ0(r) are specific for a given combination of porous material and fluid. Theoretical I(q) for monodisperse spherical pores with Percus-Yevick short-range order shows a negligible shift of qmax with increasing film thickness t because the structure factor is independent of t for Percus-Yevick systems.14 I(q)was found to increase with t up to a certain value of t for a given pore size and high porosity. Then I(q) decreases with further increases of t. In contrast, for low porosity, I(q) decreases with t for all t. The latter behavior has been observed with small angle scattering for the adsorption of contrast-matched hexane43 and dibro(43) Lin, M. Y.; Sinha, S. K.; Huang, J. S.; Abeles, B.; Johnson, J. W.; Drake, J. M.; Glinka, C. J. Mat. Res. Symp. Proc. 1990, 166, 449.

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Figure 14. Two-point probabilities (left) and size distribution of the liquid-free pore space (right) for Gelsil 50 before and after nitrogen adsorption at P/Ps ) 0.338, φ ) 0.370, and U ) 0.414. (a) Evacuated material. S2(r) from SANS data (dashed line) and Sˆ 2(r) for the reconstruct (continuous line). (b) Material exposed to nitrogen at P/Ps ) 0.338. S2(r) from SANS data (dashed line) and Sˆ 2(r) for the reconstruct (continuous line). (c) Evacuated material. BJH pore size distribution φΨ(D) (dashed line) and the PSD φΨ ˆ (D) for the empty reconstruct (continuous line). (d) Material exposed to nitrogen at P/Ps ) 0.338. Ψcorr ) βφΨcond derived as described in section 5.4 (dashed line) and the distribution φΨ ˆ (D) of the liquid-free pore space for the reconstruct (continuous line). Table 2. Morphological Parameters for Gelsil 50 Obtained from SANS, Nitrogen Adsorption Data, and Reconstruct, with the Latter Representing Average Values and Standard Deviation of 10 Realizations parameter

SANS

mean chord length hl (nm) mean chord length of pores hlpore (nm) mean chord length of the skeleton hlsolid (nm) mean pore diameter D h (nm) interface area S (m2g)

1.9 5.2 3.0

momethane44 on the random two-phase glass Vycor 7930 (φ ) 0.28, mean pore size ∼4 nm). qmax shifted from 0.24 nm-1 for the evacuated sample to 0.22 nm-1 at P/Ps ) 0.76 where 82% of the pore volume was filled by liquid dibromomethane. For P/Ps e 0.76, all γ0(r) curves agreed within the error limits given by the size of the symbols used in Figure 1 in ref 45. The mean chord length hl and the correlation length lc for Gelsil 50 are plotted as a function of the film thickness t in Figure 7. Both lengths increase with increasing t. For single particles, lc ) hl2/lh,30 and therefore lc > l, which has also been found for random two-phase media (e.g., for microporous glassy carbons34). The CLD curves A(r) for nitrogen adsorption in Gelsil 50 are shown in Figure 8. The curves for 0.054 e P/Ps e 0.465 form (44) Mitropoulos, A. Ch.; Haynes, J. M.; Richardson, R. M.; Kanellopoulos, N. K. Phys. ReV. B 1995, 52, 10035. (45) Kikkinides, E. S.; Kainourgiakis, M. E.; Stefanopoulos, K. L.; Mitropoulos, A. Ch.; Stubos, A. K.; Kanellopoulos, N. K. J. Chem. Phys. 2000, 112, 9881.

519

N2 adsorption

reconstruct

4.59 548

2.035 ( 0.013 5.526 ( 0.023 3.225 ( 0.020 4.515 ( 0.008 506.7 ( 3.300

one group with similar shapes, which is equivalent to the grouping of the corresponding γ0(r) (Figure 5). Obviously A(r f 0) * 0, although the A(r) data are restricted to r > 1 nm. Angularity in Gelsil 50 is probably due to the nearly spherical shape of the basic structural units. A TEM micrograph of Gelsil 200 with a mean pore size of 20 nm shows partially merged spherical particles,8 but the structural units in Gelsil 50 are too small to be resolved by TEM.9 Tchoubar35 found an increase in A(r f 0) due to the aggregation of silica particles. Condensation of nitrogen in the gaps between the structural units of Gelsil 50 results in leveling off the angularity, and A(r f 0) decreases with increasingP/Ps, as observed before with nitrogen adsorption on other mesoporous silicas.36 The maximum in A(r) is due to the superposition of the chord distributions of the solid/liquid phase and the liquid-free pore space. The maximum rmax is shifted to higher r with decreasing φ. Comparison with Figure 4 shows

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Figure 15. Slice with a thickness of 15 nm cut from a 3D reconstruct of Gelsil 50 with a size of 1003 nm3 (Figure 16). Table 1 shows φ and U for the P/Ps values indicated. The reconstruct of the original Gelsil 50 shows cross sections of pores (white), the internal surface of pores (dark gray), and the skeleton (gray). The nitrogen quantity present in the reconstruct at P/Ps ) 0.054 is not sufficient to cover the available interface area completely with a monolayer. The internal pore surfaces become clearly visible at P/Ps ) 0.112 because they turn green as a result of being covered with nitrogen. With increasing P/Ps, the concentration of small liquid-free spaces decreases. At P/Ps ) 0.618, bubbles with a size of ∼8 nm remain.

that this shift is mainly due to the strong increase in the mean length of the chords in the solid/liquid phase with decreasing φ. S2(r) obtained from eq 24 and calculated for the reconstructs are displayed in Figures 10, 14, and 15. 5.2. Reproducibility of the Reconstructs. To verify the reproducibility of the reconstruction procedure, we generated 10 Gelsil 50 models by means of the knowledge-based evolutionary minimization technique as described in ref 8. The model represents 1003 nm3 of the sample at a resolution of 2/3 nm per voxel (i.e., the digitized pore system is periodically embedded into a cube of 1503 voxels). Figure 9 depicts the average of the target function E that mainly involves S2 and Ψ. Figure 10 compares the average of the two-point probability Sˆ 2 of the reconstructs to S2 as derived from SANS. Analogously, Figure 11 compares the average of the geometric PSD Ψ ˆ obtained from the reconstructs to the BJHPSD Ψ. Table 2 gives the morphological parameters as derived from SANS in comparison to the corresponding averages as obtained from the reconstructed models at their lowest values

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Figure 16. Three-dimensional reconstructs with a size of 66.63 nm3 of Gelsil 50. Grey, white, and green represent the solid phase, the liquid-free space, and the nitrogen, respectively. The labyrinthine character of the pore system is evident. For details, see the legend of Figure 15.

of E. These parameters are very reproducible, with deviations of less than 8% between SANS and model data. 5.3. Estimation of the Pore Coordination Number. We estimated the coordination number Z by means of the watershed algorithm (WA), a classical tool in image processing and mathematical morphology that is used to partition a 2D image or a 3D region into nonoverlapping segments.46-49 For each voxel V, WA requires its minimum distance d(V) from the pore surface, which can be computed with the Euclidean distance transformation (EDT). For 2D images, h(V) :) -d(V) represents a 3D landscape where, metaphorically speaking, the deepest valley corresponds to the largest pore diameter. Successive rainfall leads to rising water levels in disjoint segments; thus a watershed (for a 2D image) or a separating surface (for a 3D medium) is created, and the originally disjointed segments are labeled as neighbors whenever two of them meet. Then the number of watersheds between a given segment and its neighbors is the connectivity or coordination number for this particular segment. We used EDT and WA programs as provided by ITWM Karlsruhe of (46) Shih, F. Y.; Wu, Y. Comput. Vision Image Understanding 2003, 93, 195. (47) Soille, P.; Vincent, L. Visual Commun. Image Process. 1990, 1360, 240. (48) Soille, P.; Vincent, L. IEEE Trans. Pattern Anal. Machine Intell. 1991, 13, 583. (49) Svensson, S.; Borgefors, G. Pattern Recognit. Lett. 2002, 23, 1407.

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Figure 17. Two-point probabilities for Gelsil 50 before and after exposure to nitrogen. Table 1 shows φ and U for the P/Ps values indicated. S2(r) data were derived from SANS data (dashed line) and Sˆ 2(r) from the reconstructs (continuous line).

Fraunhofer Gesellschaft.50 The averaged Z distributions of the 10 Gelsil 50 reconstructs are shown in Figure 12. The most frequent and the mean values of the distributions are Zmax ) 5.5 and Z h ) 8, respectively. A mean value of Z h perc ) 5.6 was derived by analyzing nitrogen adsorption data, in terms of a bond percolation model with heterogeneous nucleation, for a different sample of Gelsil 50 (lot 53901-18).17 It is likely that the disagreement between Z h and Z h perc is not due to a possible difference in the microstructure of both Gelsil 50 lots because the scattering curves and the nitrogen adsorption isothermes for both lots are very similar. Obviously, Z h>Z h perc because the Z distributions exhibit a long tail; this may be due to oversegmentation, a known deficiency of WA. 5.4. Reconstruction of the Spatial Nitrogen Distribution. The evolutionary method used for the generation of the material models can also be applied to reconstruct the nitrogen distribution within the mesoporous network. The main steps of this procedure are illustrated in parts a-d of Figure 13, which show slices of a thickness of 15 nm cut parallel to one of the faces of the periodical cubic model: (a) A fraction of (1 - φ) ) 0.369 of the total of 1503 digital voxels serves as the solid phase and is randomly distributed over the total volume, and the remaining part (φ ) 0.631) is reserved for the modeling of the pore space. (b) From this random starting configuration, the empty Gelsil 50 is reconstructed by means of the descriptors S2(r) and φΨ(D). (c) To bias the evolutionary reconstruction process toward capillary condensation, part of the nitrogen is prefilled into pores with diameters D e Dcond, here for a fraction of the liquid-free pore space of φ ) 0.370 at P/Ps ) 0.338 and Dcond ) 2.66 nm; the rest of the nitrogen voxels are randomly spread out within the pore system. (d) Finally, the nitrogen distribution is reconstructed by using S2(r) and Ψcorr(D) ) βφΨcond(D) as a hybrid target function in the evolutionary minimization algorithm, (50) C-library a4iL-Documentation; ITWM Karlsruhe der Fraunhofer Gesellschaft GmbH, 2004.

Figure 18. Pore size distribution φΨ(D) for evacuated Gelsil 50, size distribution of the liquid-free pore space Ψcorr in Gelsil 50 exposed to nitrogen at certain P/Ps values, and size distribution of the liquid-free pore space φΨ ˆ (D) in the reconstructs. Table 1 shows φ and U for the P/Ps values listed in the inset. (a) The agreement between the size distributions derived from experimental data and the reconstructs is reasonable for P/Ps < 0.465 (i.e., within the monolayer/multilayer adsorption regime). (b) At P/Ps g 0.465, the distribution φΨ ˆ (D) shows more small liquid-free volumes than the distribution Ψcorr(D).

whereby the capillary-condensed nitrogen is allowed to be redistributed within the whole pore system. Figure 14 shows agreement between the target descriptors and the corresponding descriptor profiles Sˆ 2 calculated from the reconstructs. The deviation of Sˆ 2 from S2 at the lowest r is due to a discretization effect; see ref 8 for details. For P/Ps ) 0.338, Figure 13d shows many essentially symmetrical liquid-free voids of diameter ∼4 ( 1 nm and voids of size ∼5 nm × 10 nm that represent cuts of pores with their axes parallel to the plane of the cut through the 3D model. The tiny liquid-free spots are sections of larger pores. Figures 15 and 16 illustrate the nitrogen distributions at five values ofP/Ps. Figure 17 demonstrates very good agreement

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Table 3. Morphological Parameters for Gelsil 50 after Nitrogen Uptake at Certain Relative Pressures P/Ps mean chord lengthsb

interface areas

mean diameter

relative pressure

fraction of liquid-free pore space

(SANS)

(reconstruct)

(SANS)

(reconstruct)

(BJH-PSD)

(reconstruct)

P/Ps

φa

S

Sˆ

hllf

ˆllf

D h

D h rec

0.000 0.054 0.112 0.338 0.465 0.618

0.631 0.486 0.450 0.370 0.298 0.159

519 442 429 357 283 149

507 440 420 352 281 157

5.19 4.70 4.48 4.43 4.49 4.55

5.52 4.68 4.53 4.50 4.58 4.38

4.59 4.19 4.18 4.27 4.50 5.30

4.53 4.11 4.11 4.10 4.18 4.49

a

From adsorption isotherm data. b Within the liquid-free pore space.

between the TPP functions S2 (as derived from SANS via eqs 21 and 24) and the respective profiles Sˆ 2 (calculated from the reconstructs). Figure 18 compares the size distributions Ψcorr obtained for the liquid-free pore space from eq 15 to the respective distributions Ψ ˆ of the reconstructs. For P/Ps e 0.465, we have Ψcorr(D) ≈ Ψ ˆ (D); see Figure 18a. Also, for this pressure range the morphological parameters calculated from SANS data and obtained from the reconstructs are in fair agreement, here to within e7% for P/Ps e 0.465 (Table 3). For 0.465 < P/Ps e 0.618 (i.e., within the hysteresis loop of the nitrogen adsorption isotherm), the curves Ψcorr are asymmetric and differ significantly from Ψ ˆ (Figure 18b). It is probable that Ψcorr does not adequately describe the size distribution of the liquid-free pore space during capillary condensation within the hysteresis loop. In particular, for this pressure range the mean value of the Ψ ˆ profile is significantly lower than the respective mean calculated from Ψcorr; considering the difference in shape of Ψ ˆ and Ψcorr, this is no surprise. In contrast, the two-point probability Sˆ 2 matches the respective target descriptor S2 very well at all pressures P/Ps and with respect to all morphological parameters inherently expressed through two-point probability. Next we inspect whether the calculated spatial nitrogen distributions are consistent with results obtained from SANS, MS of adsorption on CPGs,3 and generally accepted principles of fluid behavior in porous solids.20 We will first consider the mono-/multilayer range of adsorption. At P/Ps ) 0.054, the nitrogen quantity present in the sample is sufficient to cover 79% of the BET interface area; however, only 50% of this area is covered by nitrogen in the reconstruct. At P/Ps ) 0.112, only 60% of the interface area in the reconstruct is covered although the nitrogen quantity in the sample would suffice to cover 99%. The discrepancy indicates the failure of the evolutionary optimization technique to model the adsorbate behavior at the molecular level, which is no surprise because the statistical descriptors used here do not describe interactions between nitrogen molecules and substrate atoms. In addition, the nitrogen distributions represent statistical averages, and this implies statistical fluctuations of the nitrogen concentration on the pore surface. In contrast, MS of adsorption in atomistic models of CPGs, at low P/Ps, shows the gradual formation of a continuous adsorbate layer that is thicker on convex parts of the surface and shows bridges at constrictions in the pore system. Second, we discuss the reconstructed spatial nitrogen distribution in the multilayer adsorption and capillary-condensation regimes. As Figures 15 and 16 demonstrate, the average thickness of the adsorbed layer increases, and the number of liquid-free volumes decreases with increasing P/Ps. Clustering of bubbles is not observed on adsorption. Figure 19 shows the fraction Nfilled D (P/Ps)/N(D) of voxels in pores of diameter D filled with capillary-condensed liquid at a given relative pressure 0.338 e P/Ps e 0.618, with N(D) being the number of voxels present in pores of diameter D. Table 1 gives the Kelvin diameters Dcond

for a hemispherical meniscus (eq 1). Because of the limited resolution used in the reconstruction, approximate values D ˆ cond had to be used, as listed in the inset of Figure 19. For P/Ps ) 0.618, the use of D ˆ cond ) 5.33 instead of Dcond ) 5.44 nm gave an unreasonable result; hence we used the next possible lower value D ˆ cond ) 4.67 nm. The solid lines refer to reconstructs obtained by initially distributing all nitrogen randomly to the pore space. The broken lines belong to reconstructs achieved after prefilling all pores with D e D ˆ cond, spreading the remaining nitrogen randomly to all empty pores, and then submitting all nitrogen voxels to the minimization process. According to the BJH model, Nfilled ˆ cond (eq 1), and this D (P/Ps)/N(D) ) 1 for D < D ratio should continuously decrease with D for D > D ˆ cond in accordance with eq 2. The capillary-condensation condition is satisfied to within 10% of Nfilled D (P/Ps)/N(D) for the curves derived with prefilling, and the expected continuous decrease with increasing D is also observed. However, the difference in the results obtained with and without prefilling demonstrates that the solutions are not unique. We prefer the spatial distribution

Figure 19. Fraction of pores filled with nitrogen in the reconstruct of Gelsil 50 for certain P/Ps values. N(D) is the number of voxels filled (D) is the number of present in pores with diameter D, and NP/P s voxels in pores with diameter D filled with nitrogen at a given relative pressure. The inset shows the Kelvin diameter for a hemispherical meniscus (eq 1). According to the BJH model, fille d NP/P (D)/N(D) should be equal to 1 for D e Dcond and should s continuously decrease with D for D > Dcond. Because of the limited resolution used in the reconstruction, approximate values of Dcond were used (i.e., D ˆ cond is equal to 2.67, 3.33, and 4.67 nm instead of 2.91, 3.77, and 5.44 nm, respectively). The continuous lines represent data derived from reconstructs obtained by initially distributing nitrogen voxels randomly within the pore space of the reconstruct. The broken lines represent data obtained by forcing nitrogen into all pores with D e D ˆ cond at the given P/Ps in order to simulate capillary condensation. However, the position of these nitrogen voxels was not fixed so that these voxels could take part in the optimization process. For details, see the text.

Nitrogen Distribution in Mesoporous Gelsil 50

obtained with prefilling because the latter seems to guide the minimization process into a direction very probably taken by nature also.

6. Summary and Conclusions For a monolithic Gelsil 50 sample exposed to relative nitrogen pressures within the range of 0 e P/Ps e 0.786 at 77.7 K, the scattered intensity I(q) was measured by means of the in situ setup SANSADSO14 (Figure 6). The spatial nitrogen distribution was characterized in the multilayer adsorption and capillarycondensation regimes by statistical parameters obtained from I(q) (Figure 4, Table 3). Furthermore, correlation function γ0(r) (Figure 5), two-point correlation S2(r) (Figure 17), and chordlength distribution A(r) (Figure 8) were derived from I(q). The dependence of γ0(r) on P/Ps (Figure 5) is consistent with monolayer/multilayer formation without noteworthy capillary condensation at pressures below the lower closure point of the hysteresis loop of the isotherm and corresponds to the generally accepted model of adsorbate behavior. A marked change of γ0(r) at higher P/Ps indicates capillary condensation. A(r f 0) data indicate partial filling of angular volumes by adsorbate and capillary condensate with increasing P/Ps (Figure 8). An evolutionary minimization algorithm was employed to reconstruct digitized models of both the empty sample and the spatial nitrogen distributions within the pore system. To verify the reliability and reproducibility of the method, porosity φ, two-point probability S2(r) (Figure 10), and the volume-based pore size distribution Ψ(D) (Figure 1) were used as target descriptors for 10 reconstructions of empty Gelsil 50, each model containing 1503 voxels at a resolution of 2/3 nm per voxel. The morphological parameters calculated from the reconstructs are very reproducible, with deviations of