Simulations of Sol Gel Materials

Introduction

Silica xerogels are candidate materials for chemical sensors [1,2,3,4,5,6], drug-delivery systems [7], and novel optical [8,9,10,11] and electrochromic[12,13] applications. They are ubiquitous in chromatography [14,15,16] and catalysis [17,18,19], and are widely used in studies of gas separations [20,21] and as a support matrix for nanocluster research [22,23,19].

Aerogels are very high-porosity materials [24,25,26] used in particle detectors and as thermal insulation [27,26], in space probes (for comet-tail dust collection [28]), and in many studies of fluids confined in random media, especially helium and helium mixtures[29,30].

Silica xerogels and aerogels prepared with titanium[31,32], vanadium[33,34,35], or other metal dopants [36], or prepared from other oxides entirely[37,38,17,11,39], are promising materials for catalysts and catalyst supports[17,40,41,42] and in electrochemical applications [38,43] and solar cells [12,44].

Thin films of xerogels can be prepared by a variety of processes, including spin-coating and dip-coating [45]. Thin films are used in sensors, electronics, optics, lubrication, and other areas.

Templated materials arrived in 1992 with the first preparation of ``MCM-41'' [46], demonstrating that highly regular pore structures could be achieved on scales much larger than those present in zeolites. Recent developments in templating [47,48,49,50,51,52,53,54] and other micro-patterning technologies[55] suggest the rational design of porous media for different applications [56,54]. A wide variety of microstructures have now been prepared, including arrays of simple geometries, bicontinuous networks, and hierarchical structures [54,57].

More information on sol-gel stuff in general can be found at:

• The Sol-Gel Gateway
• A Sol Gel Technology site.

Simulations of gel processing

We are taking a multi-scale approach in this work, using conventional molecular simulation at scales up to tens of nanometers and a coarse-grained particulate model at mesoscopic scales up to a few hundred nanometers.

The preparation and properties of xerogels[58,59] and aerogels [27,26] have been comprehensively reviewed. These materials are prepared through sol-gel processing, in which precursor solutions undergo gelation, aging and drying. Xerogels are prepared by drying at subcritical solvent conditions. Liquid-vapor interfaces develop in the drying gel, and forces due to surface tension cause substantial collapse of the gel structure as liquid is removed. Aerogels are dried under supercritical (or other [60]) conditions, leading to dry gels with porosity as high as 99.9%. Consolidation, or heating at high temperatures, is used to generate densified, non-porous materials for optics and other applications.

Simulating the preparation of xerogels and aerogels involves separate treatment of gelation, aging, drying, and for non-porous materials, consolidation.

Gelation

In the gelation step, alkoxide gel precursors in aqueous solution are hydrolyzed,

and polymerize through alcohol or water producing condensations:

The gel morphology is influenced by temperature, the concentrations of each species (attention focuses on r, the water/alkoxide molar ratio, typically between 1 and 50), and especially acidity:

• Acid catalysis generally produces weakly-crosslinked gels which easily compact under drying conditions, yielding low-porosity microporous (smaller than 2 nm) xerogel structures (Figure 3a).

• Conditions of neutral to basic pH result in relatively mesoporous xerogels after drying, as rigid clusters a few nanometers across pack to form mesopores. The clusters themselves may be microporous.

• Under some conditions, base-catalyzed and two-step acid-base catalyzed gels (initial polymerization under acidic conditions and further gelation under basic conditions [61,59]) exhibit hierarchical structure and complex network topology (Figure 3c).

The initial stages of gelation, when the average cluster size is very small, are best modeled with a purely atomistic approach. Considerable effort has already gone into developing potential models for this, with convincing results [62,63,64,65,66,67,68].

Hierarchically structured gels and low-density gels cannot be directly treated with molecular models; a meso-scale approach must be used in this case. Relatively dense gels can be modeled with either atomistic simulations or coarse-grained simulations.

Aging

Simulating aging requires the use of an approach which can access long time scales. The ``activation-relaxation technique'' (ART) [69,70,71] is being implemented for this purpose. In this method the system is repeatedly moved onto saddle-points in the potential energy hypersurface (e.g., ``activation'') and then relaxed, efficiently sampling many potential minima. These methods have been successfully applied to amorphous silica [70,71] and can be implemented as an extension of a molecular dynamics code. The relaxation step can be accomplished either by numerical optimization or with molecular dynamics.

Drying

The gel drying process consists of removal of water from the gel system, with simultaneous collapse of the gel structure, under conditions of constant temperature, pressure, and humidity.

In the coarse-grained model (below) the equation of state is trivially calculable, and drying is easily modeled by choosing the solvent chemical potential to favor the vapor phase and allowing the particle positions and cell volume to slowly relax under the influence of solvent capillary forces.

At the molecular scale, we can model this process using an extension of the ``Gibbs Ensemble Monte Carlo'' technique for binary mixtures [72,73,74], where the mixture consists of water and atmosphere. The atmosphere will be modeled as a single-component gas. In this technique, two simulation cells are coupled by mass-exchange moves, in which molecules in one cell are transferred into the other cell. The volumes of the cells fluctuate independently, allowing specification of the pressure. To model constant humidity, the water content in the ``atmosphere'' cell will be controlled by periodic removal of water molecules from the simulation. This is analogous to using dehumidification in an experimental setup, and has the added benefit of requiring only a relatively small ``atmosphere'' cell.

Consolidation

Aerogels

Thin films

Experimental studies involving sol-gel nanocoating have been reviewed recently by Caruso and Antonietti[54]. The deposition of a gel of several nanometers' thickness upon a surface or nanoparticle allows one to generate novel nanostructured silica materials (via templating, below), or to modify the surface properties of the system. Simulating such processes requires the introduction of the surface or nanoparticle into the simulation cell and suitable intermolecular potentials. For deposition on planar surfaces, as in spin-coating, the concentration of the sol increases as the solvent evaporates, which can be accounted for using the drying methodology discussed above.

Templated materials

Our exploratory simulations in this area will focus on two systems with stiff and soft templates, respectively: rigid nanotubes [76], which are easily modeled for these purposes, and the quaternary ammonium surfactants used in preparation of MCM-41 [46], parameterized using the AMBER force field [77].

Meso-scale particulate model and simulations

We are investigating a coarse-grained model for sol-gel materials which replaces each cluster with a single ``gel particle'', while accounting for size variation of clusters, aggregation through condensation reactions, and solvent effects. The particle-particle interaction will be relatively short-ranged and of a shifted-center Lennard-Jones type, this approximates particle-particle interactions by the van der Waals interactions between atoms on their surfaces. Particles may also form bonds upon contact, which are described with Morse-type potentials. Particle sizes of 1-3 nm are appropriate.

The development of a solvent model suitable for drying simulations is not trivial, and will be a major methodological contribution of the proposed work. The solvent in the coarse-grained approach must (a) possesses a liquid-vapor phase diagram and reasonable interfacial properties, (b) be computationally inexpensive to solve, and (c) be sufficiently general that solvent properties and solvent-gel interactions can be fit to molecular simulation results. These requirements can be met with a lattice-gas model solved in the mean field approximation[78]. The fluid-fluid and fluid-gel interactions will be chosen to mimic atomistic potentials, truncated at a few grid spacings, and parameterized as necessary. Solvent-particle interactions are pairwise-additive, and within the mean-field approximation are simply given by a summation over forces exerted from lattice points within range of a given particle, weighted by the mean-field solution of the densities at those points. In drying simulations, constant pressure can be modeled with volume-change moves in which the simulation cell expands or contracts by one or more lattice spacings.

In strongly inhomogeneous systems, large parts of the simulation cell will be filled with bulk-like water or water vapor. These lattice points can be simply fixed at the appropriate equilibrium densities. This will substantially speed up solution of the model. Specifically, only lattice points within some threshold distance of a gel particle will be considered ``active''. As long as the pressure is either above condensation or substantially below it, this distance can be as small as several nanometers. The use of more sophisticated multi-scale ``multigrid'' techniques [79] to improve the performance of the model will be investigated and applied if possible.

Integration of molecular and meso-scale models

Integration of these two approaches requires two types of ``translation''. The first is the use of the small-scale model to parameterize the large-scale one. [The molecular model parameters could be determined, in principle, ab initio, which would introduce a third, subatomic, scale!] The second type of translation moves in the other direction - once the meso-scale model has been used to generate a structure, how can an atomic-scale description of (part of) that structure be regenerated? This fine-graining is necessary for simulations of molecular-scale processes occurring within a mesoscopic system.

Dense xerogels, which can be treated with both types of simulation, will be used to parameterize the meso-scale model. The initial stages of gelation in the molecular model can be analyzed to create an ensemble of gel particles by tabulation of the size distribution and dispersion of bonding sites. Simulations of the meso-scale model can then be initiated, and its parameters adjusted to obtain agreement of the global structural properties described above.

To reverse the process, one must replace a collection of gel particles with atomistically modeled silica. This is a non-trivial task, since it requires adding atomic-scale information to the system, rather than removing it. Two strategies will be employed. In the first, the particle configuration will be used as a template for ``carving'' the atomistic model from a block of amorphous silica, modeled separately. Such templating schemes have been effectively used for MCM-41 type materials [80], with molecular simulation used to equilibrate the unrelaxed surfaces. In the second approach, an ensemble of molecular clusters can be extracted from molecular simulations and used to replace gel particles in the coarse-grained model, matched according to size and arrangement of bonding sites. This would result in a molecular model where only inter-cluster interactions are unrelaxed; molecular dynamics can then be used to achieve mechanical stability.

Thus, in modeling the synthesis of a hierarchically structured or low-density material, we begin with molecular simulations of the gelation step. When the length-scale of aggregation in this simulation exceeds a threshold fraction of the cell size, the simulation would switch over to a coarse-grained model with equivalent statistical properties, in a much larger cell. This would be used for the remainder of gelation, aging, and drying. For characterizations by gas adsorption or surface structural analysis, a molecular model could be regenerated from the final coarse-grained configuration. For small-angle scattering or network analysis, either regenerated molecular models or meso-scale models could be used.

Bibliography

1

Y. Wei, J. G. Xu, Q. W. Feng, H. Dong, and M. D. Lin.
Encapsulation of enzymes in mesoporous host materials via the nonsurfactant-templated sol-gel process.
Mater. Lett., 44(1):6-11, 2000.

2

O. S. Wolfbeis, I. Oehme, N. Papkovskaya, and I. Klimant.
Sol-gel based glucose biosensors employing optical oxygen transducers, and a method for compensating for variable oxygen background.
Biosen. and. Bioelec., 15(1-2):69-76, 2000.

3

A. Kumar, R. Malhotra, B. D. Malhotra, and S. K. Grover.
Co-immobilization of cholesterol oxidase and horseradish peroxidase in a sol-gel film.
Anal. Chim. Acta, 414(1-2):43-50, 2000.

4

Q. Chen, G. L. Kenausis, and A. Heller.
Stability of oxidases immobilized in silica gels.
J. Am. Chem. Soc., 120(19):4582-4585, 1998.

5

M. T. Reetz.
Entrapment of biocatalysts in hydrophobic sol-gel materials for use in organic chemistry.
Adv. Mater., 9(12):943, 1997.

6

J. Lin and C. W. Brown.
Sol-gel glass as a matrix for chemical and biochemical sensing.
Trends Anal. Chem., 16(4):201-211, 1997.

7

T. K. Jain, I. Roy, T. K. De, and A. Maitra.
Nanometer silica particles encapsulating active compounds: a novel ceramic drug carrier.
J. Am. Chem. Soc., 120(43):11092-11095, 1998.

8

G. Brusatin, M. Guglielmi, P. Innocenzi, A. Martucci, and G. Scarinci.
Materials for photonic applications from sol-gel.
J. Electroceram., 4(1):151-165, 2000.

9

R. M. Almeida.
Spectroscopy and structure of sol-gel systems.
J. Sol-Gel Sci. Tech., 13(1-3):51-59, 1998.

10

R. R. A. Syms, A. S. Holmes, W. Huang, V. M. Schneider, and M. Green.
Development of the SC-RTA process for fabriation of sol-gel based silica-on-silicon integrated optic components.
J. Sol-Gel Sci. Tech., 1-3:509-516, 1998.

11

S. K. Poznyak, D. V. Talapin, and A. I. Kulak.
Structural, optical and photoelectrochemical properties of nanocrystalline TiO2-In20 composite solids and films prepared by sol-gel method.
J. Phys. Chem. B, 105(21):4816-4823, 2001.

12

A. Agrawal, J. P. Cronin, and R. Zhang.
Review of solid-state electrochromic coatings produced using sol-gel techniques.
Sol. Eng. Mater. Sol. Cells, 31(1):9-21, 1993.

13

D. R. Rosseinsky and R. J. Mortimer.
Electrochromic systems and the prospects for devices.
Adv. Mater., 13(11):783, 2001.

14

S. M. Fields.
Silica xerogel as a continuous column support for high-performance liquid chromatography.
Anal. Chem., 68(15):2709-2712, 1996.

15

M. Pursch and L. C. Sander.
Stationary phases for capillary electrochromatography.
J. Chromat. A, 887(1-2):313-326, 2000.

16

M. M. Collinson.
Sol-gel strategies for the preparation of selective materials for chemical analysis.
Crit. Rev. Anal. Chem., 29(4):289-311, 1999.

17

M. A. Cauqui and J. M. Rodriquez-Izquierdo.
Application of the sol-gel methods to catalyst preparation.
J. Non-Cryst. Solids, 147:724-738, 1992.

18

M. P. Coles, C. G. Lugmair, K. W. Terry, and T. D. Tilley.
Titania-silica materials from the molecular precursor Ti[OSi((OBu)-Bu-t)3]3: selective epoxidation catalysts.
Chem. Mater., 12(1):122-131, 2000.

19

A. G. Sault, A. Martino, J. S. Kawola, and E. Boespflug.
Novel sol-gel-based Pt nanocluster catalysts for propane dehydrogenation.
J. Catal., 191(2):474-479, 2000.

20

J. Ghassemzadeh, L. F. Xu, T. T. Tsotsis, and M. Sahimi.
Statistical mechanics and molecular simulation of adsorption in microporous materials: pillared clays and carbon molecular sieve membranes.
J. Phys. Chem. B, 104(16):3892-3905, 2000.

21

P. S. Rallabandi and D. M. Ford.
Permeation of small molecules through polymers confined in mesoporous media.
J. Membrane Sci., 171(2):239-252, 2000.

22

J. P. Carpenter, C. M. Lukehart, S. B. Milne, D. O. Henderson, R. Mu, and S. R. Stock.
Formation of crystalline nanoclusters of Ag, Cu, Os, Pd, Pt, Re, or Ru in a silica xerogel matrix from single-source molecular precursors.
Chem. Mater, 9(12):3164-3170, 1997.

23

C. A. Morris, M. L. Anderson, C. I. Merzbacher, and D. R. Rolison.
Silica sol as a nanoglue: Flexible synthesis of composite aerogels.
Science, 284:622-624, 1999.

24

G. A. Nicolaon and S. J. Teichner.
Préparation des aérogels de silice à partir d'orthosilicate de méthyle en milieu alcoolique et leurs propriétés.
Bull. Soc. Chim. France, 5:1906, 1968.

25

G. W. Scherer.
Characterization of aerogels.
Adv. Coll. Int. Sci., 76-77:321-339, 1998.

26

J. Fricke and A. Emmerling.
Aerogels - recent progress in production techniques and novel applications.
J. Sol-Gel Sci. Tech., 13:299-303, 1998.

27

J. Fricke.
Aerogels and their applications.
J. Non-Cryst. Solids, 147&148:356-362, 1992.

28

http://stardust.jpl.nasa.gov, 1999.
NASA Stardust Mission web-site.

29

S. B. Kim, J. Ma, and M. H. W. Chan.
Phase diagram of 4He-3He mixture in aerogel.
Phys. Rev. Lett., 71(14):2268-2271, 1993.

30

J. Yoon, D. Sergatskov, J. Ma, N. Mulders, and M. H. W. Chan.
Superfluid transition of 3He in ultralight aerogel.
Phys. Rev. Lett., 80(7):1461-1464, 1998.

31

Y. Han, H. Wang, H. Chen, R. Jin, and J. Deng.
Dispersed vanadium phosphorus oxide on titania-silica xerogels: highly active for selective oxidation of propane.
New. J. Chem., pages 1175-1175, 1998.

32

L. Marchese, E. Gianotti, V. Dellarocca, T. Maschmeyer, F. Rey, S. Coluccia, and J. M. Thomas.
Structure-functionality relationships of grafted Ti-MCM41 silicas. Spectroscopic and catalytic studies.
Phys. Chem. Chem. Phys., 1:585-592, 1999.

33

M. D. Curran, D. D. Pooré, and A. E. Stiegman.
Vanadia-silica sol-gel derived glass: Factors affecting homogeneity and morphology.
Chem. Mater., 10:3156-3166, 1998.

34

D. Wei, W.-T. Chueh, and G. L. Haller.
Catalytic behavior of vanadium substituted mesoporous molecular sieves.
Catalysis Today, 51:501-511, 1999.

35

X. T. Gao and I. E. Wachs.
Investigation of surface structures of supported vanadium oxide catalysts by UV-vis-NIR diffuse reflectance spectroscopy.
J. Phys. Chem. B, 104(6):1261-1268, 2000.

36

J. Shen, Q. Li, B. Zhou, J. Wang, and L. Chen.
SiO2-GeO2 binary aerogels with ultralow density.
J. Non-Cryst. Solids, 220:102-106, 1997.

37

A. C. Pierre, E. Elaloui, and G. M. Pajonk.
Comparison of the structure and porous texture of alumina gels synthesized by different methods.
Langmuir, 14:66-73, 1998.

38

J. W. Long, K. E. Swider-Lyons, R. M. Stroud, and D. R. Rolison.
Design of pore and matter architectures in manganese oxide charge-storage materials.
Elec. Solid-State Lett., 3(10):453-456, 2000.

39

W. Dong and B. Dunn.
Sol-gel synthesis and characterization of molybdenum oxide gels.
J. Non-Cryst. Solids, 225:135-140, 1998.

40

G. M. Pajonk.
Catalytic aerogels.
Catalysis Today, 35:319-337, 1997.

41

M. L. Anderson, C. A. Morris, R. M. Stroud, C. I. Merzbacher, and D. R. Rolison.
Colloidal gold aerogels: Preparation, properties and characterization.
Langmuir, 15:674-681, 1999.

42

S. Yoda, K. Ohtake, Y. Takebayashi, T. Sugeta, T. Sako, and T. Sato.
Preparation of titania-impregnated silica aerogels and their application to removal of benzene in air.
J. Mater. Chem., 10:2151-2156, 2000.

43

J. R. Jurado, E. Chinarro, and M. T. Colomer.
Ceramic conductors for electrochemical cell applications: new perspectives in materials and synthesis.
Solid State Ionics, 135(1-4):365-372, 2000.

44

D. G. Chen.
Anti-reflection (AR) coatings made by sol-gel processes: A review.
Sol. Eng. Mater. Sol. Cells, 68(3-4):313-336, 2001.

45

C. J. Brinker, A. J. Hurd, P. R. Schunk, G. C. Frye, and C. S. Ashley.
Review of sol-gel thin film formation.
J. Non-Cryst. Solids, 147:424-436, 1992.

46

J. S. Beck, J. C. Vartuli, W. J. Roth, M. E. Leonowicz, C. T. Kresge, K. D. Schmitt, C. T. W. Chu, D. H. Olson, E. W. Sheppard, S. B. McCullen, J. B. Higgens, and J. L. Schlenker.
A new family of mesoporous molecular-sieves prepared with liquid-crystal templates.
J. Am. Chem. Soc., 114(27):10834-10843, 1992.

47

R. A. Caruso, M. Giersig, F. Willig, and M. Antonietti.
Porous `coral-like' TiO2 structures produced by templating polymer gels.
Langmuir, 14(22):6333-6336, 1998.

48

P. D. Yand, D. Y. Zhao, D. I.Margolese, B. F. Chmelka, and G. D. Stucky.
Generalized syntheses of large-pore mesoporous metal oxides with semicrystalline frameworks.
Nature, 396:152-155, 1998.

49

B. T. Holland, C. F. Blanford, and A. Stein.
Synthesis of macroporous minerals with highly ordered three-dimensional arrays of spheroidal voids.
Science, 281:538-540, 1998.

50

G. S. Attard, J. C. Glyde, and C. G. Göltner.
Liquid-crystalline phases as templates for the synthesis of mesoporous silica.
Nature, 378:366-368, 1995.

51

C. G. Göltner and M. Antonietti.
Mesoporous materials by templating of liquid crystalline phases.
Adv. Mater., 9(5):431-436, 1997.

52

V. Z.-H. Chan, J. Hoffman, V. Y. Lee, H. Iatrou, A. Avgeropoulos, N. Hadjichristidis, R. D. Miller, and E. L. Thomas.
Ordered bicontinuous nanporous and nanorelief ceramic films from self assembling polymer precursors.
Science, 286:1716-1719, 1999.

53

H. Wakayama, H. Itahara, N. Tatsuda, S. Inagaki, and Y. Fukushima.
Nanoporous metal oxides synthesized by the nanoscale casting process using supercritical fluids.
Chem. Mater., 13(7):2392-2396, 2001.

54

R. A. Caruso and M. Antonietti.
Sol-gel nanocoating: An approach to the preparation of structured materials.
Chem. Mater., ASAP:on-line, 2001.

55

N. Kroger, R. Deutzmann, and M. Sumper.
Polycationic peptides from diatom biosilica that direct silica nanosphere formation.
Science, 286:1129-1132, 1999.

56

T. J. Barton, L. M. Bull, W. G Klemperer, D. A. Loy, B. McEnaney, M. Misono, P. A. Monson, G. Pez, G. W. Scherer, J. C. Vartuli, and O. M. Yaghi.
Tailored porous materials.
Chem. Mater., 11:2633-2656, 1999.

57

T. R. Pauly and T. J. Pinnavaia.
Pore size modification of mesoporous HMS molecular sieve silicas with wormhole framework structures.
Chem. Mater, 13(3):987-993, 2001.

58

C. J. Brinker and G. W. Scherer.
Sol-Gel Science.
Acad. Press, San Diego, 1990.

59

C. J. Brinker.
Structure of sol-gel-derived glasses.
In Glass Science and Technology, volume 4A, pages 169-230. Academic Press, Inc., Boston, 1990.

60

M.-A. Einarsrud.
Light gels by conventional drying.
J. Non-Cryst. Solids, 225:1-7, 1998.

61

C. J. Brinker, W. D. Drotning, and G. W. Scherer.
A comparison between the densification kinetics of colloidal and polymeric silica gels.
In C. J. Brinker, D. E. Clark, and D. R. Ulrich, editors, Better Ceramics Through Chemistry, volume 32 of Symposia Proceedings, pages 25-32, New York, 1984. Materials Research Society, North-Holland.

62

B. P. Feuston and S. H. Garofalini.
Empirical three-body potential for vitreous silica.
J. Chem. Phys., 89(9):5818-5824, 1988.

63

B. P. Feuston and S. H. Garofalini.
Oligomerization in silica sols.
J. Phys. Chem., 94:5351-5356, 1990.

64

B. P. Feuston and S. H. Garofalini.
Onset of polymerization in silica sols.
Chem. Phys. Lett., 170(2-3):264-270, 1990.

65

B. P. Feuston and S. H. Garofalini.
Water-induced relaxation of the vitreous silica surface.
J. Appl. Phys., 68(9):4830-4836, 1990.

66

S. H. Garofalini and G. Martin.
Molecular simulations of the polymerization of silicic acid molecules and network formation.
J. Phys. Chem., 98:1311-1316, 1994.

67

K. Yamahara and K. Okazaki.
Molecular dynamics simulation of the structural development in sol-gel process for silica systems.
Fluid Phase Eq., 144:449-459, 1998.

68

E. B. Webb III and S. H. Garofalini.
Relaxation of silica glass surfaces before and after stress modification in a wet and dry atmosphere: molecular dynamics simulations.
J. Non-Cryst. Solids, 226:47-57, 1998.

69

G. T. Barkema and N. Mousseau.
Event-based relaxation of continuous disordered systems.
Phys. Rev. Lett., 77(21):4358-4361, 1996.

70

N. Mousseau and G. T. Barkema.
Traveling through potential energy landscapes of disordered materials: the activation-relaxation technique.
Phys. Rev. E, 57(2):2419-2424, 1998.

71

N. Mousseau, G. T. Barkema, and S. W. de Leeuw.
Elementary mechanisms governing the dynamics of silica.
J. Chem. Phys., 112(2):960-964, 2000.

72

A. Z. Panagiotopoulos.
Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble.
Mol. Phys., 61(4):813-826, 1987.

73

A. Z. Panagiotopoulos.
Adsorption and capillary condensation of fluids in cylindrical pores by Monte Carlo simulation in the Gibbs ensemble.
Mol. Phys., 62(3):701-719, 1987.

74

A. Z. Panagiotopoulos, N. Quirke, M. Stapleton, and D. J. Tildesley.
Phase equilibria by simulation in the Gibbs ensemble - alternative derivation, generalization and application to mixture and membrane equilibria.
Mol. Phys., 63(4):527-545, 1988.

75

P. Jiang, J. F. Bertone, and V. L. Colvin.
A lost-wax approach to monodisperse colloids and their crystals.
Science, 291:453-457, 2001.

76

B. C.Satishkumar, A. Govindaraj, E. M. Vogl, L. Basumallick, and C. N. R. Rao.
Oxide nanotubes prepared using carbon nanotubes as templates.
J. Mater. Res., 12(3):604-606, 1997.

77

P. K. Weiner and P. A. Kollman.
Amber: Assisted model building with energy refinement. A general program for model molecules and their interactions.
J. Computat. Chem., 2:287-299, 1981.

78

E. Kierlik, M. L. Rosinberg, G. Tarjus, and P. Viot.
Equilibrium and out-of-equilibrium (hysteretic) behavior of fluids in disordered porous materials: Theoretical predictions.
Phys. Chem. Chem. Phys., 3:1201-1206, 2001.

79

E. L. Briggs, D. J. Sullivan, and J. Bernholc.
Real-space multigrid-based approach to large-scale electronic structure calculations.
Phys. Rev. B, 54:14362-14375, 1996.

80

B. P. Feuston and J. B. Higgins.
Model structures for MCM-41 materials: A Molecular Dynamics simulation.
J. Phys. Chem., 98:4459-4462, 1994.