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Simulated Annealing as a Method for the Determination of the Spatial Distribution of a Condensable Adsorbate in Mesoporous Materials M. E. Kainourgiakis,*,† E. S. Kikkinides,‡ G. Ch. Charalambopoulou,† and A. K. Stubos† National Center for Scientific Research “Demokritos”, 15310 Ag. Paraskevi Attikis, Greece; and Center for Research and Technology Hellas, Chemical Process Engineering Research Institute, Sixth km. Charilaou- Thermi Road, 57001 Thermi, Thessaloniki, Greece Received October 29, 2002. In Final Form: January 28, 2003
The simulated annealing algorithm is employed for the determination of the spatial distribution of a condensable adsorbate in reconstructed digital domains of mesoporous materials. The optimum configuration for a given degree of saturation is derived assuming that in equilibrium the total interfacial free energy reaches a global minimum value. The suggested approach is applied to reconstructed matrixes of Vycor and random sphere packs. Furthermore, the effective diffusivities of inert gases in the Knudsen regime are calculated in the resulting “dry” and “wet” images with a step-by-step random walk method. The predicted gas relative permeability curves are in satisfactory agreement with experimental results, confirming that the suggested combination of digital reconstruction and simulated annealing is a useful tool for the determination of the spatial distribution of a condensable sorbed phase in mesoporous solids.
Introduction Adsorption systems based on mesoporous solids are involved in a wide range of industrial processes, such as gas separation and purification, oil recovery, heterogeneous catalysis, and so forth. The analysis of the adsorption isotherms obtained from this specific type of materials is closely related to the concept of capillary condensation,1 which describes a gas-liquid phase transition originating from the confined geometry of the pore space. Capillary condensation is said to occur when, during adsorption from a vapor phase, the thin layer building up on the pore walls reaches a metastable state to finally collapse and condense at a certain pressure related to the value of the pore radius.1 Determining the distribution of the condensed adsorbate in the porous space is of primary importance, since it can greatly contribute not only to the prediction of various properties of the particular system but also to our understanding of the overall sorption process. Even though there is a well-established theory for the description of adsorption mechanisms in mesoporous media,1 the adequate representation of the three-phase system (vaporadsorbate-solid matrix) forming during the sorption process is extremely difficult due to the complicated morphology that is induced by porosity. Significant efforts have been devoted to the study of the particular physical behavior exhibited by an adsorbate due to the confinement and the geometrical disorder of the porous matrix.2 The gas-liquid transition within pores has been extensively studied by both theoretical and simulation methods, as * To whom correspondence should be addressed. Telephone: +30-210-6503447. Fax: +30-210-6525004. E-mail: kainourg@ chem.demokritos.gr. † National Center for Scientific Research “Demokritos”. ‡ Chemical Process Engineering Research Institute. (1) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (2) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573.
thoroughly reviewed by Gelb et al.2 A big part of these studies concern fluids confined in single pores of simple and regular shape (usually slitlike or cylindrical). Even though the results for idealized pore geometries are of great scientific interest, several complicating factors such as interpore correlation effects, variations in pore geometry, networking of pores, and so forth, need to be considered. This is why the quantitative description of the complex porous microstructure has been the subject of continuous research. Starting from rather simplified representations of the internal structure of porous media through pore networks,3 the evolution of experimental and numerical techniques allowed the development of two different modeling approaches that are currently used for the production of binary images of the biphasic porous matrix: (a) the statistical reconstruction methods,4-8 where the binary array reflects a number of statistical properties of the actual biphasic medium, and (b) the process-based methods,9-11 where the computational procedure tries to imitate the physical processes that take place during the formation of the medium. The binary domains resulting from either approach can be used for the simulation of either dynamic processes, such as diffusion12,13 and permeation,5,14 or equilibrium properties, such as sorption.15-19 (3) Nicholson, D.; Petropoulos, J. H. J. Chem. Soc., Faraday Trans. 1 1984, 80, 1069. (4) Quiblier, J. A. J. Colloid Interface Sci. 1986, 98, 84. (5) Adler, P. M. Porous Media: Geometry and Transports; Butterworth: London, 1992. (6) Torquato, S. Random Heterogeneous Materials: Microstructure and Macroscopic Properties; Springer-Verlag: New York, 2002. (7) Levitz, P.; Pasquier, V.; Cousin, I. In Characterization of Porous Solids IV; McEnaney, B., Mays, T. J., Rouqe´rol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., Eds.; Bath, 1996; p 135. (8) Crossley, P. A.; Schwartz, L. M.; Banavar, J. R. Appl. Phys. Lett. 1991, 59, 3553. (9) Bryant, S. L.; Cade, C. A.; Mellor, D. W. AAPG Bull. 1993, 77, 1338. (10) Bakke, S.; Øren, P. E. SPE J. 1997, 2, 136. (11) Øren, P. E.; Bakke, S.; Arntzen, O. J. SPE J. 1998, 3, 324.
10.1021/la026766p CCC: $25.00 © 2003 American Chemical Society Published on Web 03/05/2003
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Both the aforementioned reconstruction approaches have been successfully employed in previous studies12,20 by the present authors in which the structures of two typical mesoporous materials were reproduced in a sufficiently accurate manner. More specifically, a ballistic deposition process of spheres was employed as a processbased representation method of alumina mesoporous membranes made by compaction of spherical monosize alumina particles, while a stochastic reconstruction technique generated 3-D images of Vycor porous glass. In the present work, these reconstructed alumina and Vycor domains are used, in an attempt to determine the corresponding concentration profile of a condensable adsorbate for a given degree of saturation Vs (fraction of pore space occupied by the adsorbate), assuming that in equilibrium the total interfacial free energy is minimal. A simulated annealing method is employed for the optimization procedure. In a further step, the suggested approach is validated by calculating the transport properties of the reconstructed domains, namely the effective diffusivity of an inert gas such as He, by combining the mean square displacement method with a standard random walk process, and the gas relative permeability, PR, by dividing the computed effective diffusivity at each degree of saturation by that computed for each of the dry materials. Numerical Methodology Representation of Porous StructuresDry Vycor and Alumina Geometry. The spatial distribution of matter in a porous medium can be typically represented by the phase function Z(x), defined as follows:
Z(x) )
{
1, if x belongs to the pore space 0, otherwise
(1)
where x is the position vector from an arbitrary origin. Due to the disordered nature of porous media, Z(x) can be considered as a stochastic process, characterized by its statistical properties. The porosity and the autocorrelation function Rz(u) can be defined by the statistical averages:5,21
) 〈Z(x)〉 Rz(u) )
〈(Z(x) - )(Z(x + u) - )〉 - 2
(2a) (2b)
Note that 〈 〉 indicates spatial average. For an isotropic medium, Rz(u) becomes one-dimensional, as it is only a function of u ) |u|. A reliable reconstruction of a porous medium in three dimensions should possess the same correlation properties as those determined in a single two-dimensional section, (12) Kainourgiakis, M. E.; Kikkinides, E. S.; Stubos, A. K.; Kanellopoulos, N. K. J. Chem. Phys. 1999, 111, 2735. (13) Kainourgiakis, M. E.; Kikkinides, E. S.; Steriotis, Th. A.; Stubos, A. K.; Tzevelekos, K. P.; Kanellopoulos, N. K. J. Colloid Interface Sci. 2000, 231, 158. (14) Bekri, S.; Adler, P. M. Int. J. Muliphase Flow 2002, 28, 665. (15) Gelb, L. D.; Gubbins, K. E. Langmuir 1998, 14, 2097. (16) Woo, H. J.; Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 7472. (17) Sarkisov, L.; Monson, P. A. Langmuir 2001, 17, 7600. (18) Sarkisov, L.; Monson, P. A. Phys. Rev. E 2001, 65, 011202. (19) Kierlik, E.; Monson, P. A.; Rosinberg, M. L.; Sarkisov, L.; Tarjus, G. Phys. Rev. Lett. 2001, 87, 055701. (20) Kainourgiakis, M. E.; Kikkinides, E. S.; Stubos, A. K. J. Porous Mater. 2002, 9, 141-154. (21) Berryman, J. G. J. Appl. Phys. 1985, 57, 2374.
properly reflected by the various moments of the phase function.20 In most cases, knowledge of the first two moments, namely porosity and autocorrelation function, is considered sufficient for this purpose. In this respect, a standard stochastic reconstruction technique was employed to generate digitized three-dimensional porous images of Vycor porous glass, with the same porosity ( ) 0.28) and two-point autocorrelation function as those of the actual material. The two-point autocorrelation function used as input was taken from the literature.12 A linear filter produced by the given two-point autocorrelation function was applied to a random uncorrelated Gaussian field with zero mean and unit variance, resulting in a correlated Gaussian field. The correlated field was then passed through a nonlinear discretization filter to produce the target phase function Z(x) (i.e. the stochastically reconstructed sample).20 The size of the produced binary domains was 128 × 128 × 128, and six different realizations were considered in order to calculate the diffusion coefficients. The pixel size was equal to 1.5 nm. Furthermore, the reconstructed domains shared the same internal surface area with the actual material, since this structural property could be determined from the slope of the autocorrelation function at zero distance.22 In addition, for the reconstruction of the structure of an alumina membrane,20 fabricated by the compaction of alumina microspheres with diameter 20 nm, an algorithm simulating the random packing of hard spheres has been applied,13 based on the random sequential deposition of non-overlapping spherical particles.23 In such a ballistic deposition, the packing rule differs from other methods in that the spheres position themselves under the influence of a unidirectional (vertical) force, rather than toward a center of attraction. The basic idea of the algorithm is as follows: Balls are dropped sequentially from a random point well above the simulation box. Each time N “test” balls are dropped, only the one whose final position is lowest (provided that no overlapping with the spheres in the stack occurs) becomes a part of the stack. If N is large enough (N > 105), then random sphere packs with the same structural properties found by the more rigorous deposition algorithms are recovered. Following this process, five different sphere packs were generated and discrete domains with size equal to 128 × 128 × 128 were produced. The number of spheres in each pack was approximately 103, while the porosity of the produced domains was in the range 0.4-0.42, yielding a sphere radius of approximately 6.7 pixels. Given that for the preparation of the alumina membrane the diameter of the spheres used was 20 nm, the pixel size was equal to 1.5 nm. Determination of the Spatial Distribution of Adsorbate in Reconstructed Domains. During an adsorption experiment involving a mesoporous solid (S), a condensable gas is progressively introduced in the porous matrix and as a result a layer of adsorbate (L) is initially building up on the pore walls. When condensation occurs, all the pores with radii smaller than a critical value are progressively blocked, and the adsorbate is in equilibrium with its vapor (V). To obtain a realistic configuration of the condensed phase in a reconstructed mesoporous domain for a given degree of pore filling (saturation), the arrangement of the fluid phases can be optimized by assuming that in equilibrium the total interfacial free energy, GS, associated with the multiphase (22) Debye, P.; Anderson, H. R., Jr.; Brumberger, H. J. Appl. Phys. 1957, 28, 679. (23) Vold, M. J. J. Phys. Chem. 1960, 64, 1616.
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system, is minimal,24 according to
GS )
∑i Aiγi
(3)
where Ai is the interfacial area of i (equal to 1 square pixel in our simulations) and γi is the interfacial free energy of i. In the present case, the total interfacial energy has contributions from three different interfaces: (a) condensed phase (L)-solid (S), (b) vapor (V)-solid (S), and (c) condensed phase (L)-vapor (V). The corresponding γi components always obtain positive values and for complete wetting of the solid surface (contact angle equal to 0°) cancel according to the Young-Dupre´ equation:
γSL ) γSV - γLV
(4)
The actual distribution of phase L is represented by the arrangement of the fluid pixels that minimizes the total interfacial energy. In the present work, a simulated annealing (SA) algorithm is employed for the determination of the GS minimum. SA is a powerful Monte Carlo technique for solving optimization problems, mainly characterized by the ability of escaping local minima, since it incorporates a probabilistic criterion for accepting or rejecting new solutions.25 Due to its implementation simplicity and efficiency, its use has expanded to numerous diverse applications spanning from printed circuit board design25 to the reconstruction of porous structures26,27 and the determination of optimal soil sampling strategies for precision agriculture research.28 Simulated annealing was originally proposed in 1983 by Kirkpatrick, Gelatt, and Vecchi,25 who considered the use of the Metropolis algorithm to solve some of the typical combinatorial problems appearing in the field of circuit design. To achieve this, they established an analogy between the annealing process in solids, the behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature, and the optimization problem of finding the global minimum of a multiparameter objective function. In the pertinent method, a randomly generated perturbation of the current configuration of the system under study is applied so that a trial configuration is obtained. Let Ec and Et denote the energy level of the current and trial configurations, respectively. If Ec g Et, then a lower energy level has been reached, the trial configuration is unconditionally accepted and becomes the current configuration. On the other hand, if Ec < Et, then the trial configuration is accepted with a probability given by
(
P(∆E) ) exp -
)
∆E kBT
(5)
where ∆E ) Et - Ec, kB is the Boltzmann constant, and T is the temperature (or an arbitrary analogue of it, used only to symbolically represent the degree of randomness in the spatial distribution of the system phases). This step prevents the system from being trapped in a local lowest-energy state. After a sufficient number of iterations, the system approaches equilibrium, where the free energy reaches its minimum value. By gradually decreasing T (24) Knight, R.; Chapman, A.; Knoll, M. J. Appl. Phys. 1990, 68, 994. (25) Kirkpatrick, S.; Gelatt, C. D.; Vecchi, M. P. Science 1983, 200, 671. (26) Torquato, S. J. Chem. Phys. 1999, 111, 8832. (27) Talukdar, M. S.; Torsaeter, O.; Ioannidis, M. A. J. Colloid Interface Sci. 2002, 248, 419. (28) Van Groenigen, J. W.; Gandah, M.; Bouma, J. Soil Sci. Soc. Am. J. 2000, 64, 1674.
and repeating the simulation process (using every time as the initial configuration the one found as the equilibrium state for the previous T value), new lower energy levels become achievable. The process is considered complete when, despite the change in T, the number of accepted changes in different configurations becomes lower than a prespecified value. In the present work, the objective function to be minimized is the total interfacial energy, GS. To find the equilibrium distribution of liquid and vapor for a certain Vs, the required number of liquid elements, L, is first randomly selected and thus an initial state of the system is defined corresponding to a reference total energy value, G0. Progressively, the positions of one site in the pore space occupied by the L phase and of one V element are swapped, preserving the specified saturation. Let us consider the n-th trial of this process. If G ) (Gtrial - Gn-1) < 0, the position interchange is allowed and Gn ) Gtrial. In the case that Gtrial > Gn-1, the new configuration is accepted with a probability given by exp(-G/Gref), where Gref is a control parameter analogous to temperature T, having the same units as the objective function GS. The calculations are repeated until no change in Gn is observed. Then, the value of Gref is decreased by 1% and the procedure is repeated. As GS approaches its minimum and Gref is decreased, the number of acceptable configurations also decreases. This is used as a criterion for the termination of the calculation. When the ratio of the number of acceptable moves to the number of trials is lower than 10-5, the calculation is finished. Simulation of Knudsen Diffusion in Reconstructed Domains. Knudsen diffusion is simulated by monitoring the trajectories of a statistically sufficient number (∼104) of identical pointlike inert particles injected in the void space of the porous matrix and undergoing off-lattice random walk. The total number of time steps for every particle is 106, long enough to feel the structural details of the porous domain. The direction of the motion of each particle changes when it hits the molecularly rough solid or liquid surface, where it gets reflected diffusely according to cosine law. The orientationally averaged diffusivity of pointlike inert tracers in the binary domains is calculated from the mean-square displacement 〈r2〉, from the wellknown Einstein relation:
〈r2〉 tf∞ 6t
D ) lim
(6)
where t is the travel time of the particles. The displacement is monitored throughout the distance, s, traveled by the particles, assuming that they move at a constant speed equal to the mean thermal speed,12,29,30 〈uth〉 ) (8RT/πM)1/2. Using the exact solution for an infinitely long tube of diameter le, in the Knudsen regime, Dk ) 1/3le〈uth〉, and substituting t ) s/〈uth〉, eq 6 takes the dimensionless form
〈r2/le2〉 D ) lim Dk s/lef∞ 2s/le
(7)
In addition, the incorporation of porosity in diffusivity yields the effective diffusivity Deff ) D, also known as permeability, P, for the case of inert gases. Results and Discussion The numerical scheme based on simulated annealing, as described above, determines the optimum condensate (29) Tomadakis, M. M.; Sotirchos, S. V. AIChE J. 1993, 39, 397. (30) Burganos, V. N. J. Chem. Phys. 1998, 109, 6772.
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Figure 1. 3-D simulated images of a single partially filled rectangular pore through the course of the SA simulation: (a) starting random configuration; (b) final configuration (for illustrational purposes only half of the pore is shown).
configuration, taking into account the capillary condensation phenomenon that progresses with increased saturation, and therefore assumes that the whole fluid quantity, occupying a certain portion of the pore space for a given degree of saturation, exhibits completely liquidlike properties. This condition does not constrain the ability of the suggested approach to imitate the physical system. Figure 1 illustrates the picture obtained from the simulation procedure for a single rectangular pore. Initially, a certain fraction of the void space is occupied by condensate distributed in a random mode (Figure 1a) and the system is evolving toward the minimum of total interfacial energy. The initial random distribution of the condensate clearly converges to the formation of a bulk liquid phase, separated from the gas phase by a concave meniscus (Figure 1b). Likewise, three-dimensional “wet” Vycor and alumina domains, obtained during the course of the SA are shown in Figure 2. It is clear that when equilibrium is reached (starting from the random fluid configurations of Figure 2a and c), the adsorbate forms clusters localized in narrow pores (Figure 2b and d), in accordance with the basic principles of adsorption in mesoporous media. The adsorbate accumulation is further elucidated by Figure 2e and f, where the solid phase has been subtracted from the alumina images. The spatial distribution of the condensate and the subsequent partial blockage of the porous network have a strong impact on the dynamic properties of the medium. Experimentally, this can be investigated by partially filling the porous matrix with a stationary condensed phase and subsequently measuring the permeability of a second nonadsorbable gas, which does not condense in the pores, at least under the specific conditions. This type of experiment is known as gas relative permeability,31 and it is usually performed at low pressures where the mean free path of the gas molecules is much greater than the characteristic pore length. In this way, intermolecular collisions are rare and the transport process is determined primarily by the collisions between the molecules and the solid walls. This mechanism resides within the Knudsen regime, and thus permeability reduces to Knudsen diffusivity. In this respect, the binary matrixes of the reconstructed Vycor and alumina served as a basis for calculating by a step-by-step random walk procedure, the permeability of an inert gas. The pertinent computations on the stochastically reconstructed domains of dry Vycor and on the (31) Ash, R.; Barrer, R. M.; Pope, C. G. Proc. R. Soc. 1963, A271, 1.
Figure 2. Spatial distribution of a condensed adsorbate in 3-D reconstructed Vycor (a and b) and alumina (c-f) images (Vs ) 0.6) as produced by the SA simulation. The left column refers to the corresponding starting random configurations, and the right, to the final optimized ones.
ballistic depositions of the dry alumina membrane resulted in a very good agreement with corresponding experimental values of Knudsen diffusivity. The computed effective Knudsen diffusivity of He, for the case of a dry Vycor binary domain with porosity ) 0.28 and pixel size 1.5 nm, has been found equal to 8.4 × 10-4 cm2/s, while a value of 8.5 × 10-4 cm2/s was obtained from He permeability measurements32,33 at 298 K. For the case of a dry alumina random sphere pack domain with porosity ) 0.41 and sphere diameter 20 nm, the computations resulted in a Knudsen diffusivity equal to 7.7 × 10-3 cm2/s, matching the experimental value of 8.0 × 10-3 cm2/s that was deduced from experiments13 on a γ-alumina membrane at 308 K. It must be noted that the calculated Knudsen diffusivity can be compared with that determined by Kainourgiakis et al.,13 where the random sphere packs were off-lattice represented. The observed very good agreement shows that the discretization of the domain is accurate enough and the noise imposed by the rough “stairlike” (see Figure 2d) structure is negligible. Helium diffusivity was also calculated for the case of binary domains representing the wet Vycor and alumina structures for various degrees of saturation, Vs. The effect of the nature of the adsorbate on its spatial distribution within the mesoporous matrix and therefore on the transport properties of the system was investigated by (32) Satterfield, C. N.; Sherwood, T. K. The Role of Diffusion in Catalysis; Addison-Wesley: Reading, MA, 1963. (33) Makri, P. K.; Romanos, G. E.; Steriotis, Th. A.; Kanellopoulos, N. K.; Mitropoulos, A. Ch. J. Colloid Interface Sci. 1998, 206, 605.
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Figure 3. Simulated and experimental gas relative permeability curves of He on Vycor porous glass partially filled with a condensable adsorbate phase. The solid line is used only as a guide.
considering different sets of interfacial energies γi. For each of the resulting systems gas relative permeability, PR, was determined by normalizing the effective He diffusivity computed for a certain Vs value by the one for the dry material. The simulation results are compared against experimental data obtained from the measurement of He permeability in Vycor preadsorbed with CH2Br2 and γ-alumina preadsorbed with CCl4. For the case of the reconstructed Vycor domains, a good agreement between simulation and experiment33 occurs (Figure 3), demonstrating that the suggested approach can reproduce adequately the concentration profile of a condensable adsorbate in this specific type of material. The simulation results manage to capture not only the overall decay behavior but also specific features of the experimental relative permeability curve (e.g. prediction of the percolation threshold, VSC). In addition, it is interesting to point out that despite the significant variation of γi components, always under the condition of complete wetting (γSV > γSL), the computed relative permeability values fall upon a single curve. This demonstrates that the spatial distribution of the condensate is not affected significantly by the actual values of γi involved, in agreement with the notion that when a wetting phase is present under equilibrium it must always occupy the same positions, regardless of the absolute values of the individual interfacial energies. Similar conclusions can be drawn when considering the case of the alumina membrane (Figure 4). Again the simulation results manage to predict the trend of the experimental relative permeability curve.34 The slight deviation between the absolute values of computed and experimental PR can be attributed to presumable structural defects (stratification) of the membrane system used in the corresponding measurements.34 Since the fabrication process of this specific membrane involved the segmental addition of alumina powder and the uniaxial (34) Papadopoulos, G. Study of adsorption, diffusion and gas relative permeability in mesoporous alumina membranes, in relation to their porous and macroscopic structure (in Greek). Ph.D. Thesis, University of Athens, 1993.
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Figure 4. Simulated and experimental gas relative permeability of He on alumina partially filled with a condensable adsorbate phase. The solid line is used only as a guide.
compression of the subsequent layers, one of the main experimental difficulties was to attain an efficient control of the porosity homogeneity and of the maintenance of the particle shape. Therefore, it is quite likely that the resulting sphere pack is characterized by a heterogeneous (layered) structure that gives rise to the observed slight discrepancies in Figure 4. Finally, it should be noted that mean-field solutions of the lattice gas model19 may provide both the isotherm and the spatial distribution of the adsorbate within the reconstructed pore space accounting for nonzero density of the vapor phase. However, the calculation of effective Knudsen diffusivities requires the existence of either full (solid or condensate) or empty (gas) voxels in the digitized representation of the spatial distribution of phases. This means that a thresholding procedure should have been used if the spatial distribution of the phases were to be provided as a real function by the mean-field lattice gas model solution. Conclusions The binary reconstruction of mesoporous domains in combination with the SA algorithm offers a sufficient tool for the determination of the spatial distribution of a condensable adsorbate in mesoporous solids, for a given degree of saturation. This is supported first by the visual inspection of the obtained three-dimensional “wet” images, showing that the capillary condensation can be reliably reproduced, as well as by the corresponding calculated transport properties (effective Knudsen diffusivities). It is furthermore shown that when the capillary condensation is the dominant adsorption mechanism, the spatial distribution of the adsorbate depends mainly on the morphology of the porous structure and not on the nature of the solid-adsorbate interaction reflected by the actual values of the individual interfacial energies. According to this, the gas relative permeability curve for a specific mesoporous medium might exhibit a universal behavior regardless of the type of the adsorbate and therefore can be used for the characterization of the pore space topology. LA026766P
