Application of Percolation Theory to the Drainage of Liquid Nitrogen

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Application of Percolation Theory to the Drainage of Liquid Nitrogen from Mesoporous Silica Xerogel Gelsil 50 Ernst Hoinkis*,† and Barbara Ro¨hl-Kuhn‡ Hahn-Meitner-Institut Berlin GmbH, Strukturforschung I, Glienickerstr. 100, D-14109 Berlin, Germany and Bundesanstalt fu¨ r Materialforschung und -pru¨ fung, D-12489 Berlin, Germany Received March 3, 2005. In Final Form: May 30, 2005

Previous in situ small-angle neutron scattering studies of nitrogen adsorption and desorption at 78 K on the mesoporous silica xerogels Gelsil 50 and Gelsil 75 revealed the formation of ramified clusters of vapor-filled pores on desorption, which is characteristic for a percolation process. In the present work, we check whether the adsorption/desorption isotherm data for a monolithic sample of Gelsil 50 can be analyzed in terms of a bond-percolation model. Three powder samples were studied too. Percolation probability data are presented and the effects of heterogeneous nucleation, finite size, and surface clusters on drainage from Gelsil 50 are addressed. The mean coordination number was derived. The results of the analysis are discussed with respect to recent theoretical work for interactions of fluids with complex pore systems. The monolithic sample and a powder sample were characterized by small-angle neutron scattering data.

1. Introduction The pore space of many adsorbents and catalysts consists of three-dimensional (3D) interconnected randomly meandering cavities of varying shape and crosssection. The adsorption/desorption isotherms for such mesoporous complex pore systems exhibit hysteresis: desorption from capillary condensed liquid-filled pores occurs at a lower relative pressure P/Ps than adsorption. Wall and Brown1 interpreted hysteresis in terms of percolation theory:2-12 capillary condensation occurs without restraint anywhere within the pore system if the external P/Ps is higher than *P/Ps for condensation. Transport of adsorbate occurs either through the vapor or the liquid. On reduction of P/Ps, pores near the sample boundary drain first, and then drainage proceeds within a narrow pressure range sequentially along the path of least resistance. A pore drains if the external P/Ps is lower than *P/Ps and if the pore is connected either to the sample boundary or to a vapor-filled cavity. Drainage of a larger * Author to whom correspondence should be addressed. E-mail: [email protected]. Telephone: +49 30 8062-2803. † Hahn-Meitner-Institut Berlin GmbH. ‡ Bundesanstalt fu ¨ r Materialforschung und -pru¨fung. (1) Wall, G. C.; Brown, R. J. C. J. Colloid Interface Sci. 1981, 82, 141. (2) Broadbent, S. R.; Hammersley, J. M. Proc. Cambridge Philos. Soc. 1957, 53, 629. (3) Kirkpatrick, S. In Ill-Condensed Matter; Balian, R., Maynard, R., Toulouse, G., Eds.; North-Holland: Amsterdam, 1979; p 321-388. (4) Wilkinson, D.; Willemsen, J. F. J. Phys. A: Math. Gen. 1983, 16, 3365. (5) Neimark, A. V. Colloid J. 1984, 46, 813. Translated from Kolloidn. Zh. 1984, 46, 927. (6) Zhdanov, V. P.; Fenelonov, V. B.; Efremov, D. K. J. Colloid Interface Sci. 1987, 120, 218. (7) Mason, G. Proc. R. Soc. London, Ser. A 1988, 415, 453. (8) Stauffer, D.; Aharony, A. Introduction to Percolation Theory; Taylor & Francis: London, 1992. (9) Zhdanov, V. P. In Advances in Catalysis; Eley, D. D., Pines, H., Weisz, P. B., Eds.; Academic Press: San Diego, 1993; Vol. 39, pp 1-50. (10) Sahimi, M. Flow and Transport in Porous Media and Fractured Rock; VCH: Weinheim, 1995. (11) Yortsos, Y. C. In Experimental Methods in the Physical Sciences; Celotta, R., Lucatorto, T., Eds.; Academic Press: San Diego, 1999; pp 69-113. (12) Lenormand, R. In Experimental Methods in the Physical Sciences; Celotta, R., Lucatorto, T., Eds.; Academic Press: San Diego, 1999; pp 43-66.

liquid-filled pore is retarded if the latter is connected to the sample boundary via smaller liquid-filled pores only. This pore blocking is the basic assumption of percolation theory applied to desorption. A review of earlier work related to pore blocking may be found in ref 13. Singlepore hysteresis is neglected in the percolation approach. Drainage of liquid is equivalent to the invasion of vapor into the liquid-filled pore system.4 Sahimi reviewed drainage experiments with etched networks, and two- and three-dimensional unconsolidated glass beads, which support the invasion percolation model.10 According to percolation theory, the vapor-filled space forms a sample spanning fractal cluster at the knee of the desorption isotherm. A fractal dimension, Dv ) 2.53, was obtained from Monte Carlo simulations of random and invasion percolation in infinite 3D lattices.14 Provided the experiments cover suitable length scales, Dv may be derived from light and neutron scattering. The intensity scattered from volume fractals satisfies I(q) ∼ q-Dv within 1/ξ < q < 1/a, where q is the modulus of the scattering vector, ξ is the maximum correlation length over which the cluster is fractal, and a is the smallest pore size.15-17 Page et al.18,19 studied ad- and desorption of n-hexane on Vycor by light scattering. On desorption, they observed a fractal cluster with Dv ) 2.6. Small-angle neutron scattering (SANS) studies of adsorption and desorption of water on Vycor20,21 and other systems have been reviewed in ref 22. (13) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2002, 18, 9830. (14) Sheppard, A. P.; Knackstedt, M. A.; Pinczewski, W. V.; Sahimi, M. J. Phys. A: Math. Gen. 1999, 32, L521. (15) Sinha, S. K.; Freltoft, T.; Kjems, J. In Kinetics of Aggregation and Gelation; Family, F., Landau, D. P., Eds.; Elsevier Sci. Publ., 1984; pp 87-90. (16) Teixeira, J. In: On Growth and Form, Fractal and Nonfractal Patterns in Physics; Stanley, H. E., Ostrowsky, N., Eds.; Martinus Nijhoff: Dordrecht, 1986; pp 145-162 (17) Teixeira, J. J. Appl. Cryst. 1988, 21, 781. (18) Page, J. H.; Liu, J.; Abeles, B.; Deckman, H. W.; Weitz, D. A. Phys. Rev. Lett. 1993, 71, 1216. (19) Page, J. H.; Liu, J.; Abeles, B.; Herbolzheimer, E.; Deckman, E. W.; Weitz, D. A. Phys. Rev. E 1995, 52, 2763. (20) Li, J.-C.; Ross, E. K.; Benham, M. J. J. Appl. Crystallogr. 1991, 24, 794. (21) Li, J.-C.; Ross, D. K.; Howe, L. D.; Stefanopoulos, K. L.; Fairclough, J. P. A.; Heenan, R.; Ibel, K. Phys. Rev. B 1994, 49, 5911. (22) Hoinkis, E. Part. Part. Syst. Charact. 2004, 21, 80.

10.1021/la050580j CCC: $30.25 © 2005 American Chemical Society Published on Web 07/06/2005

Mesoporous Silica Xerogel Gelsil 50

The SANS experiments reveal the formation of fractal clusters on desorption, but not on adsorption. The Dv data were found to be in the range 1.51-1.79, i.e., lower than the theoretical value 2.53, which has been attributed18,19 to the short length scales of ∼1-50 nm probed by SANS. The ramification of the cluster may then be effected by short-range correlations in the pore structure. Pore blocking may also be described as the inability of filled pores deep inside the material to cavitate although the capillary condensed liquid that is thermodynamically unstable because of a negative hydrostatic pressure.23 The concept of pore blocking in single inkbottle pores has recently been questioned. Several theoretical studies revealed that the emptying mechanism of inkbottle pores depends on pore width, temperature, and the ratio of solid/fluid to fluid/fluid interaction strength. When the necks are narrow, the cavity may empty because of spontaneous cavitation, while the necks remain liquid filled.13,24-26 The pore blocking concept has also been rejected for desorption from complex pore systems. Model calculations of ad- and desorption isotherms by means of molecular simulation and density functional theory (DFT) suggest that hysteresis is due to multiple metastable states, which are induced by the disorder of the solid matrix.27-29 The metastable state model has recently been supported by calculations30 of ad- and desorption isotherms by means of DFT for a large (1923 nm3) fcc lattice model of Vycor with a porosity of 30%. The adsorption/desorption isotherms show triangular hysteresis loops as observed experimentally. Similar results were obtained from DFT calculations and molecular simulations of fluid adsorption in a lattice model of a xerogel with 67.8% porosity and in a glass with 50% porosity.31 Molecular simulations of adsorption and desorption of Xe from a Vycor model with a mean pore size of 5 nm and 30% porosity revealed that desorption proceeds without the formation of a sample spanning cluster.32 Recent molecular simulations and DFT calculations of fluid desorption indicate bubble formation on reduction of P/Ps in models of mesoporous solids with complex pore systems and porosities of 75%33 and 25%.23 Under certain conditions, the bubbles grow self-similar, coalesce, and finally extend over the whole pore system. The fractal dimension of the vapor-filled space was found to be Dv ) 2.84. Drainage proceeded like invasion percolation, although drainage was not controlled by pore blocking. The results obtained for complex pore systems may be summarized as follows: (i) A considerable number of molecular simulations and DFT studies of fluid desorption from complex pore systems reject the percolation approach and support the metastable states model.27,28,30-32 (ii) There (23) Woo, H.-J.; Porcheron, F.; Monson, P. A. Langmuir 2004, 20, 4743. (24) Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 7600. (25) Vishnyakov, A.; Neimark, A. V. Langmuir 2003, 19, 3240. (26) Libby, B.; Monson, P. A. Langmuir 2004, 20, 4289. (27) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573. (28) Pikunic, J.; Latoskie, C. M.; Gubbins, K. E. In Handbook of Porous Solids; Schu¨th, F., Sing, K. S. W., Weitkamp, J., Eds.; Wiley-VCH: Weinheim, 2002; pp182-236. (29) Wallacher, D.; Ku¨nzner, N.; Kovalev, D.; Knorr, N.; Knorr, K. Phys. Rev. Lett. 2004, 92, 195704-1. (30) Woo, H.-J.; Sarkisov, I.; Monson, P. A. Langmuir 2001, 17, 7472. (31) Sarkisov, L.; Monson, P. A. Phys. Rev. E 2001, 65, 011202 (32) Gelb, L. D.; Gubbins, K. E. In Fundamentals of Adsorption 7; Kaneko, K., Kanoh, H., Hanzawa, Y., Eds.; International Adsorption Society; IK International: Chiba City, Japan, 2002; pp 333-340 (33) Rosinberg, M. L.; Kierlik, E.; Tarjus, G. Europhys. Lett. 2003, 62, 377.

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is clear experimental evidence for percolation controlled drainage in Vycor18,19,34 and macroporous agglomerates.10,35 (iii) According to two recent molecular simulations and DFT studies, drainage from a complex pore system may proceed like invasion percolation, although drainage is not controlled by pore blocking.23,33 Obviously the characterization of complex pore systems requires further work and the combination of theoretical concepts with different experimental methods. If desorption from a complex pore system is in fact a percolation process, the pore size distribution (PSD) should be estimated from the adsorption data because the PSD estimated from the desorption branch would be too narrow. This conclusion is of some importance for routine PSD determinations by use of the Barrett-Joyner-Halenda method (BJH)36 because the standard practice according to ASTM37 and DIN38 is to use the desorption data for the calculation of the BJH-PSD. Previously, we have studied adsorption/desorption of nitrogen at 78 K on the silica xerogels Gelsil 75 and Gelsil 50 for the first time with in situ SANS. We concluded that desorption was an invasion percolation process.22,39 The goal of the present work is to find out whether adsorption/ desorption isotherms for Gelsil 50 can be analyzed in terms of the bond-percolation model40 in order to confirm or to reject our conclusion on the basis of the in situ SANS experiments. For this purpose, we have measured nitrogen adsorption/desorption isotherms at 77 K for one monolithic and three powder samples of Gelsil 50. The data set for the hysteresis loop of a given isotherm consists of 90 data points in order to facilitate the calculation of precise percolation probability data F as function of the drained pore fraction f. The measurement of one isotherm took 4 days. The monolith and one powder sample have been further characterized by SANS. We address the effects of particle size, surface clusters, and heterogeneous nucleation on the percolation probability data. 2. Characterization of Gelsil 50 by SANS and Nitrogen Adsorption/Desorption Isotherm Data Gelsil 50 is a monolithic, transparent, rigid, high-purity silica xerogel with interconnected mesopores. The preparation of Gelsils by a sol/gel process has been described in ref 41. The last step of production is densification at a temperature of ∼1173 K. A Gelsil 50 sample with a nominal pore size of 5 nm (product code A050-100-050, lot 53901-18) was obtained from Geltech Inc., 3267 Progress Dr., Orlando, FL 32837 USA. The sample was shipped evacuated and was used as received. Thickness and diameter of the cylindrical sample were 4.64 mm and 9.72 mm, respectively. The bulk density was Fbulk ) 0.907 g cm-3 and the porosity φ ) 0.675. A SANS spectrum was measured by using the V4 instrument of the Berlin (34) Schechter, R. S.; Wade, W. H.; Wingrave, J. A. J. Colloid Interface Sci. 1977, 59, 7. (35) Shaw, T. M. Phys. Rev. Lett. 1987, 59, 1671. (36) Barrett, E. P.; Joyner, L.; Halenda, P. P. J. Am. Chem. Soc. 1951, 73, 373. (37) Standard Practice for Calculation of Pore Size Distributions of Catalysts from Nitrogen Desorption Isotherms; ASTM D 4641-88; American Society for Testing and Materials: Philadelphia, 1988. (38) Bestimmung der Porengro¨ ssenverteilung und der spezifischen Oberfla¨ che mesoporo¨ ser Feststoffe durch Stickstoffadsorption, Verfahren nach Barrett, Joyner und Halenda (BJH); DIN 66134; Deutsches Institut fuer Normung eV.: Berlin, 1998. (39) Hoinkis, E.; Ro¨hl-Kuhn, B. In Fundamentals of Adsorption 7; Kaneko, K., Kanoh, H., Hanzawa, Y., Eds.; International Adsorption Society; IK International: Chiba-City, Japan, 2002; pp 601-607 (40) Seaton, N. A. Chem. Eng. Sci. 1991, 46, 1895-1909. (41) Nogues, J.-L.; Moreshead, W. V. J. Non-Cryst. Solids 1990, 121, 136.

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Figure 2. Porod plot for scattering from the monolithic Gelsil 50 sample. The background B is the slope of the plot q4I(q) vs q4.q4(I(q) - B) approaches a constant value, which indicates that the corrected data satisfy Porod’s law.

Figure 1. Logarithmic plot of the small-angle neutron scattering intensity I(q) for a cylindrical sample of Gelsil 50 (d ) 9.7 mm, h ) 4.6 mm) (b) and for a powder prepared from the cylindrical sample. The mean grain size of the powder is 0.16 mm. The Porod scattering I(q) ∼ q-4 observed for the powder at the lowest q is due to scattering from the external surface of the grains. The slight deviation from the q-4 dependence at high q is due to the background, see Figure 2.

Neutron Scattering Center at HMI. Sample/detector distances were 1.0, 4.0, and 15.85 m and the wavelength was λ ) 0.605 nm, corresponding to a q-range of 0.04-3.5 nm-1; q ) (4π/λ) sin Θ, where 2Θ is the scattering angle. The diameter of the incident beam was 2.5 mm and the transmission Tr ) 0.35. The raw data were treated by standard procedures.42 The coherent macroscopic scattering cross-section is abbreviated here as intensity I(q). The scattering curve depicted in Figure 1 shows a weak maximum at q ≈ 0.35 nm-1 and a power law decay for q > 1 nm-1, in agreement with a scattering spectrum measured previously for a Gelsil 50 monolith with a thickness of 2 mm (product code A050-060-020-K, lot 72401).43 For q > 2 nm-1, the I(q) data deviate from power law scattering. This is due to incoherent scattering as well as short-range spatial density fluctuations within the silica skeleton. The corresponding background B ) 0.0363 cm-1 is the slope of the plot q4I(q) versus q4 shown in Figure 2. The corrected (I(q) - B) data satisfy the asymptotic Porod law,44 which indicates a smooth pore surface.

CPorod ) lim q4I(q) qf∞

(1)

(42) Keiderling, U. Physica B 1997, 234, 1111. (43) Eschricht, N.; Hoinkis, E.; Ma¨dler, F.; Schubert-Bischoff, P. In Studies in Surface Science and Catalysis; Rodriguez-Reinoso, F., McEnaney, B., Rouquerol, J., Unger, K., Eds.; Elsevier Science: Amsterdam, 2002; pp 355-362. (44) Porod, G. Kolloid-Z. 1951, 83, 124.

Figure 3. Plot of q2( I(q) - B) vs q. For the calculation of the invariant Q, the integrand in eq 2 is approximated by a power law series regression of order 10 within the interval 0.04 e q(nm-1) e 2.

The Porod constant is CPorod ) 4.45 cm-1 nm-4. The invariant44 is

Q ) F∆ +

∫ab q2 (I(q) - B) dq + CPorod/b

(2)

where a ) 0.04 nm-1 and b ) 2 nm-1 are the lower limits of the experimental q-range and the q-range satisfying Porod’s law, respectively. Use of Q and CPorod for the calculation of the geometric parameters minimizes the effects of multiple scattering due to the low transmission45 of the monolith. Figure 3 shows q2 (I(q) - B) versus q. The integrand was approximated by a power series regression of order 10 within a e q e b. The value of the integral is 4.63 cm-1 nm-3. The integral between 0 and a was approximated by F∆ ) 6.4 × 10-4 cm-1 nm-3. So Q ) 6.86 (45) Schelten, J.; Schmatz, W. J. Appl. Crystallogr. 1980, 13, 385.


cm-1 nm-3. The mean chord length is ml ) (4/π) Q/CPorod ) 1.96 nm. The mean value of chords in pores is ml pore ) ml /(1 - φ) ) 6.0 nm, and the mean value of chords in the silica skeleton is ml solid ) ml /φ ) 2.9 nm. The specific surface area, SSANS ) 4φ(1 - φ)103/ml Fbulk ) 493 m2 g-1, is in reasonable agreement with the BET surface area SBET ) 534 m2 g-1, see below. After the SANS measurement, the original monolith was cut into two equal parts by use of a diamond wire saw because the original did not fit into the ASAP 2010 sample container. The sorption isotherm for nitrogen at 77.4 K was measured by using the ASAP 2010 (Micromeritics Corp., Norcross, USA) after degassing the sample at 250 °C for several hours until the residual gas pressure was less than 0.2 Pa. After the measurement of the adsorption isotherm for this halfmonolith, both parts of the original monolith were gently crushed by using a mortar and pestle. The grains were suspended in alcohol, and the fine particles (e0.01 mm) were removed by decanting three times. The remaining grains (denoted powder 1) showed the typical appearance of crushed glass. The grain size was determined from photomicrographs. The nitrogen adsorption isotherm was measured as described before. Then the powder was transferred into a silica cell, the latter was heated for 2 h at 235 °C in air, the cell was closed by use of a foil, and a SANS spectrum was measured, which is also shown in Figure 1. Sample/detector distances were 1.0 and 4.0 m with λ ) 0.605 nm, and 4 and 12 m with λ ) 1.275 nm. Diameter of incident beam, sample thickness, and transmission were 10.0 mm, 1 mm, and Tr ) 0.89. The I(q) data within the peak range and the data at higher q are similar to the data for the monolith, which shows that the microstructure of Gelsil 50 was not affected by crushing the monolith. The I(q) data at lower q for powder 1 deviate from the data for the monolith because of Porod scattering from the grain surfaces at the lowest q. Therefore, the calculation of the invariant would not give a reliable result. The microstructure of Gelsil 50 has been studied in ref 43. In short, the weak maximum at q ) 0.35 nm-1 (see Figure 1) indicates some short-range order, or in terms of chord lengths, certain chord lengths are more frequent than others. The shape of I(q) for the monolith at q < 0.35 nm-1 indicates that other short-range correlations are absent. The correlation function γ0(r) describes the microstructure and was calculated in ref 43. For r > 15 nm, γ0(r) ) exp(-r/ml ), which indicates a completely random, isotropic two-phase system. Correspondingly, the 3D reconstruct of Gelsil 50 consists of a highly interconnected pore network. The mean grain size of powder 1 was 0.16 mm, and the size range was 0.02-0.3 mm with a maximum at 0.15 mm. Powder 2 with a mean particle size of 0.026 mm (range of 0.01-0.08 mm) was prepared by grinding powder 1. Subsequently, the adsorption isotherm was measured, and powder 3 was prepared. The mean particle size of the latter is 0.008 mm (range of 0.002-0.025 mm). In addition to the particle sizes given above, powders 2 and 3 contained very fine particles (∼1 µm). These amounted to about 10% of the whole. The quantity of the powders was too small for a separation by, e.g., sedimentation, to be performed. Decanting could not be applied because the size of all particles of powders 2 and 3 was too small. Nevertheless, we calculated the percolation probability F for powders 2 and 3 in order to compare the data to the F data for powder 1 and the monolith. However, we did not quantitatively interpret the F data for powders 2 and 3. The nitrogen

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Figure 4. Nitrogen adsorption/desorption isotherm data at 77.4 K for the hysteresis loop. The mean particle size is listed in Table 1. Table 1. Mean Grain Size and Geometrical Parameters Derived from Nitrogen Adsorption Isotherm Data sample

mean grain size (mm)

half-monolith r ) 4.85; h ) 4.6 powder 1 0.16 powder 2 0.026 powder 3 0.008

SBET 4Vpore/ Vpore (cm3/g) CBET (m2/g) SBET (nm) 0.744 0.726 0.678 0.665

69.3 58.7 62.7 65.1

534 534 513 501

5.6 5.4 5.3 5.3

adsorption isotherms V(P/Ps) are shown in Figure 4 for P/Ps > 0.4. The data in the range 0 < P/Ps < 0.4 show the sigmoidal shape characteristic for multilayer adsorption in mesopores. The test for micropores by means of the Rs-method46 was negative. Saturation of the pore systems occurs at P/Ps ) 0.85. Diminution of particle size lead to a decrease in the pore volume and to intergranular condensation in powders 2 and 3. The knee of the desorption branch of the isotherm for powders 2 and 3 is less sharp than the knee observed for powder 1 and the monolith. The geometrical parameter derived from the isotherm data are listed in Table 1. The BET plots are linear within 0.05 e P/Ps e 0.3 with a correlation coefficient of 0.99998. Accordingly, the error of SBET is ≈ 1 m2 g-1, and the error of CBET is negligible. So the changes observed for SBET and CBET are systematic. The decrease in CBET may be due to the destruction of high-energy adsorption sites because of the cracking of the grains. SBET decreases with fragmentation because, in a mesoporous medium, most of the interface area is located within the pores. The moderate steepness of the adsorption branch indicates a relatively wide PSD. The distribution of pore volume V as function of pore diameter δ, Y(δ) ) -dV(δ)/dδ, was calculated according to BJH36 from the adsorption branch of the hysteresis loop by using the (46) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd. ed.; Academic Press: London, 1982; Chapter 2.

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concentration bfc is defined as the concentration at which an infinite cluster of interconnected bonds appears in an infinite lattice.The percolation probability, often called accessibility, is defined by F ) perc,bNopen/ bN with perc,bNopen equal to the number of open bonds belonging to the percolating cluster. F ) 0 for f < bfc, F > 0 for f > bfc, F ) 1 for f ) 1, and F/f ) perc,bNopen/ bNopen. For bond percolation in regular and random 3D lattices, empirically1,9,10 Z bfc ≈ 1.5. A randomized lattice may have the same percolation threshold bfc as a regular lattice.50,51 Lopez-Ramon et al.52 derived an expression that represents approximate values F for an infinite 3D network of regular and random bonds.

F ≈ (1.314u0.41 + 3.153u - 3.48u2 + 1.433u3)/Z with u ) Z f - 1.494 (5) F ) 0 for f < 1.494/Z

Figure 5. The pore size distributions as estimated with the BJH method36 from the adsorption branch of the isotherm. A hemispherical meniscus was assumed.

standard software provided by Micromeritics Corp. The expression47

t ) 0.354 (-5.0/ln(P/Ps )1/3 nm

(3)

was used to correct the Kelvin equation

P/Ps ) exp[-2γVm cos θ/RT (r - t)]

(4)

r is the radius of a cylindrical pore, γ is the surface tension of liquid adsorbate, Vm is the liquid molar volume, and the contact angle θ ≈ 0. The size distributions are shown in Figure 5. The pore size distributions reflect the adsorption isotherm data and, therefore, change with fragmentation. 3. Theoretical Background Seaton et al.40,48 assumed that all pore volume is associated with bonds. We calculate the percolation probability F according to Seaton’s approach and then apply an expression derived by Parlar and Yortsos49 to analyze the F data in terms of percolation with heterogeneous nucleation in a Bethe lattice. bN, bNopen, and bNclosed denote the number of all bonds, of open bonds, and of closed bonds, respectively. The probability that a bond is open is f ) bNopen/ bN. The connectivity Z of a regular lattice is the number of bonds leaving each site, Z ) bN/ sN, where s N is the number of sites. In a random lattice, Z may vary from site to site, and an experimentally determined value of Z represents the arithmetic average. The critical (47) Halsey, G. D. In Advances in Catalysis; Frankenstein, W. G., Komarewsky, V. I., Rideal, E. K., Eds.; Academic Press: New York, 1952; Vol. IV, pp 259-269 (48) Liu, H.; Zhang, L.; Seaton, N. A. Chem. Eng. Sci. 1992, 4393. (49) Parlar, M.; Yortsos, Y. C. J. Colloid Interface Sci. 1989, 132, 425.

and F ) f for f > 2.7/Z.

The spatial distribution of a fluid upon draining from a porous solid has also been approximated by a Cayley tree, commonly termed a Bethe lattice.7,8,53,54 The critical concentration is8 Bethefc ) 1/(Z - 1). Bonds in a Bethe lattice are branching without reconnections, in contrast to a complex real pore system. However, there are four arguments for the use of a Bethe lattice for an approximative description of drainage from interconnected pore systems: (i) The shapes of F(f) curves for Bethe lattices and regular lattices are quite similar. (ii) Randomization of a simple cubic lattice and a Bethe lattice may result in about equal F(f) data for both types of lattices.50 (iii) Closed loops are not important for drainage because, in terms of percolation theory, an avalanche of vapor-filled space is assumed to proceed from the boundary into the sample. (iv) The connectivity of a Bethe lattice can be selected such that its percolation threshold matches that of a 3D lattice; bfc for a simple cubic lattice (Z ) 6) and a Bethe lattice with Z ) 5 differ by only 0.1%. The conductivity of both lattice types is equal.10 Parlar and Yortsos49 derived an expression for F for bond percolation in an infinite Bethe lattice:

F ) f - (1 - fnuc) f [(1 - f )/(1 - x)]2(Z-1)

(6)

where fnuc is the fraction of bonds undergoing a liquid to vapor transition via heterogeneous nucleation and x is the solution of eq 7 for given values of Z, f, and fnuc.

x (1 - x)Z-2 ) (1 - fnuc) f (1 - f )Z-2 with 0 < x < 1/(Z - 1) (7) Vapor bubble formation is assumed to commence in pits with a size in the order of defects in the pore wall. The likelihood of heterogeneous nucleation is small at P/Ps near the onset of desorption and increases as P/Ps decreases. Model calculations of Parlar and Yortsos49 show that fnuc < 0.05. At bfc, a phase consisting of finite clusters is transformed into a phase in which an infinite cluster is present and the cluster size becomes the dominant length scale of the (50) Larson, R. G.; Scriven, L. E.; Davis, H. T. Chem. Eng. Sci. 1981, 36, 57. (51) Yanuka, M. J. Colloid Interface Sci. 1989, 127, 35. (52) Lopez-Ramon, M. V.; Jagiello, J.; Bandozs, T. J.; Seaton, N. A. Langmuir 1997, 13, 4435. (53) Mason, G. In Studies in Surface Science and Catalysis, Characterization of Porous Solids; Unger, K. K., Rouquerol, J., Sing, K. S. W., Kral, H., Eds.; Elsevier: Amsterdam, 1988; Vol. 39, pp 323-332. (54) Ball, P. C.; Evans, R. Langmuir 1989, 5, 714.


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system. F near bfc depends on the lattice size L, which is known as the finite-size effect.3,40 According to Wall and Brown1

(Vs - Vd) ) (Vs - Va) F/ f

(8)

P/Ps given, Va is the volume of nitrogen taken up by the pore system on adsorption, and Vd is the quantity of nitrogen remaining in the sample on desorption. F/ f refers also to the given P/Ps value. The number of open bonds b Nopen is identified with the number of pores that are vapor filled plus those pores that contain liquid in a metastable state. The ratio F/ f delays desorption because perc,bNopen ) 0 for f < fc. f can be calculated40,48 from the number distribution of pores ψ(δ). For cylindrical pores of length λ, and assuming δ and λ to be uncorrelated

ψ(δ) ) Y(δ)/(π/4)δ2λ *P/Ps

f)

∫δ*∞ ψ(δ) dδ/∫0∞ ψ(δ) dδ

(9) (10)

*P/Ps in eq 10 is the relative pressure at which nitrogen condenses in a cylindrical pore of diameter δ*. Pores with diameter δ* and connected directly to the sample boundary via empty pores drain at P/Ps < *P/Ps. Other pores with diameter δ* may be blocked. The integral in the nominator is equal to the number of drained pores plus those pores that contain capillary condensed liquid at P/Ps < *P/Ps. With known *P/Psf, the percolation probability F can be calculated with eq 8. 4. Percolation Analysis of Nitrogen Drainage from Gelsil 50 and Results f was calculated from eq 10 with δ* ) 2 nm, which is about the lower limit of the applicability of the BJH procedure. Equations 3 and 4 relate δ* ) 2r to *P/Ps. The δ* values were 3.3, 3.5, 3.7, ..., 6.7 nm corresponding to *P/Ps equal to 0.40, 0.43, 0.455, ..., 0.688, respectively. We obtained eighteen *P/Psf data. Below, we will write f instead of *P/Psf for convenience. A polynomial regression of 2nd grade to the f data was used to calculate f data for equidistant P/Ps values (∆(P/Ps) ) 0.01) within 0.40 e P/Ps e 0.79, which allowed for a convenient calculation of F data. For the calculation of F/f according to eq 8, one needs extrapolated Vs(P/Ps) data. The slope of Vs(P/Ps) reflects the decrease in the density of the capillary condensed liquid19,34 and reflects the evaporation from pores with direct connection to the sample boundary. A regression line for the quantity V in the range 0.8 e P/Ps e 0.997 was used to calculate Vs(P/Ps) data for powder 1 and the half-monolith within the range 0.4 e P/Ps e 0.79. Intergranular condensation occurs with powders 2 and 3 at P/Ps > 0.8, but is negligible at P/Ps < 0.8. The regression lines for powder 2 and 3 were calculated for 0.65 e P/Ps e 0.8. The measured Va data were approximated by a 5th grade power series regression function. The deviation between measured and predicted Va values was 100. For a comparison of theoretical F(f) data with experimental F(f) data, the particle size needs to be expressed in units of the mean pore size.40 The pore volume is 0.57 cm3 in 1 cm3 of the Gelsil 50 sample. We assume

7372


cylindrical pores of 5 nm diameter and 10 nm length. The latter was taken from a transmission electron microscopy image.43 The number of such pores is ∼3 × 1018/cm3, which corresponds to ∼105 pores/mm on one side of a hypothetical cubical sample. The mean particle size expressed in units of pore size is L ≈ 103 for powder 3. On the basis of the mean particle size, the powders 1, 2, and 3 and the halfmonolith should be equivalent to an infinite system. However, powders 2 and 3 contain about 10% very fine grains e0.001 mm, and these contribute via the finitesize effect to the measured F(f) for f < 0.22. Another contribution, surfF, to the measured F(f) for f < 0.22 comes from the evaporation from pores connected directly to the particle boundary. According to Monte Carlo simulations,48 surf F contributes significantly to F(f) for lattices with L ) 20 and L ) 60. However, surfF is negligible for much larger 3D samples with a high ratio of outer surface to volume, such as the half-monolith, but surfF is not negligible for powders 2 and 3. In short, the finite-size effect and desorption from surface clusters is negligible for the monolith and is approximately negligible for powder 1 with a mean grain size of 0.16 mm. The ramified clusters observed by in situ SANS adsorption/desorption experiments22,39 could also be formed via cavitation and the self-similar growth of bubbles that coalesce and finally extend over the whole pore system without percolation.23,33 However, the good fit of the percolation model with heterogeneous nucleation to F(f) for the half-monolith suggests that desorption of nitrogen from Gelsil 50 is likely controlled by a percolation process, which is in accordance with our previous conclusion.22,39 The connectivity derived from the percolation probability data was found to be Z ) 5.6 for a Bethe lattice model, which corresponds to Z ) 6.5 for a 3D lattice model. These Z values are near Z ) 6 for a simple cubic lattice. Z ) 6 has been assumed5 for mesoporous solids in order to calculate the neck size distribution from percolation probability data. Equation 10 is based on the assumptions that the pores are cylindrical and that diameter and length are uncorrelated. Assuming cylindrical pore shape has proved to be


suitable for the estimation of the pore size distribution of randomly interconnected mesopores by the BJH method.27 Diameter and length of pores in Gelsil 50 are uncorrelated at least for length scales >15 nm because Gelsil 50 scatters in the corresponding q range like a completely random two-phase system. There are certainly more complicated models of pore systems. However, on the basis of the evidence presented here, it seems to be justified to interpret the experimental results by using a simple bond-percolation model with heterogeneous nucleation. 6. Summary and Conclusions Desorption of nitrogen at 77.4 K from a Gelsil 50 halfmonolith is quantitatively described by a percolation process with heterogeneous nucleation. A connectivity (or mean coordination number) Z ≈ 6 was derived from the analysis of the percolation probability F(f). The experimental F(f) data for powders with mean grain sizes of 0.008 mm and 0.026 mm plus some still finer particles are significantly higher than the data for the half-monolith and those for a powder with a mean grain size of 0.16 mm. The increase in F(f) with decreasing particle size is attributed to the finite-size effect and to the evaporation from pores at the particle boundary of the finest particles. For complex pore systems such as silica xerogels, the BJH pore size distribution should be determined from the adsorption isotherm data. In situ ultra SANS nitrogen adsorption experiments at 77 K with mesoporous silicas would be useful because the percolating cluster could be observed at length scales up to several µm and it could be checked whether the observed fractal dimension is about equal to the theoretical value (2.53) for percolation. Acknowledgment. The authors would like to thank Mrs. A. Zimathies for performing the adsorption experiments, P. Schubert-Bischoff for preparing the photomicrographs, U. Keiderling for critical comments on an earlier draft, N. Eschricht and F. Ma¨dler for discussions, and F. Mezei for his continuous support of this work. LA050580J

Application of Percolation Theory to the Drainage of Liquid Nitrogen

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