Small-angle neutron-scattering investigation of long-range correlations in silica aerogels: Simulations and experiments

Deep reinforcement learning for microstructural optimisation of silica aerogels

Silica aerogels are being extensively studied for aerospace and transportation applications due to their diverse multifunctional properties. While their microstructural features dictate their thermal, mechanical, and acoustic properties, their accurate characterisation remains challenging due to their nanoporous morphology and the stochastic nature of gelation. In this work, a deep reinforcement learning (DRL) framework is presented to optimise silica aerogel microstructures modelled with the diffusion-limited cluster-cluster aggregation (DLCA) algorithm. For faster computations, two environments consisting of DLCA surrogate models are tested with the DRL framework for inverse microstructure design. The DRL framework is shown to effectively optimise the microstructure morphology, wherein the error of the material properties achieved is dependent upon the complexity of the environment. However, in all cases, with adequate training of the DRL agent, material microstructures with desired properties can be achieved by the framework. Thus, the methodology provides a resource-efficient means to design aerogels, offering computational advantages over experimental iterations or direct numerical solutions.

Orbital-flop transition of superfluid 3He in anisotropic silica aerogel

Superfluid 3He is a paradigm for odd-parity Cooper pairing, ranging from neutron stars to uranium-based superconducting compounds. Recently it has been shown that 3He, imbibed in anisotropic silica aerogel with either positive or negative strain, preferentially selects either the chiral A-phase or the time-reversal-symmetric B-phase. This control over basic order parameter symmetry provides a useful model for understanding imperfect unconventional superconductors. For both phases, the orbital quantization axis is fixed by the direction of strain. Unexpectedly, at a specific temperature Tx, the orbital axis flops by 90∘, but in reverse order for A and B-phases. Aided by diffusion limited cluster aggregation simulations of anisotropic aerogel and small angle X-ray measurements, we are able to classify these aerogels as either "planar" and "nematic" concluding that the orbital-flop is caused by competition between short and long range structures in these aerogels.

The effect of synthesis conditions and process parameters on aerogel properties

Aerogels are remarkable nanoporous materials with unique properties such as low density, high porosity, high specific surface area, and interconnected pore networks. In addition, their ability to be synthesized from various precursors such as inorganics, organics, or hybrid, and the tunability of their properties make them very attractive for many applications such as adsorption, thermal insulation, catalysts, tissue engineering, and drug delivery. The physical and chemical properties and pore structure of aerogels are crucial in determining their application areas. Moreover, it is possible to tailor the aerogel properties to meet the specific requirements of each application. This review presents a comprehensive review of synthesis conditions and process parameters in tailoring aerogel properties. The effective parameters from the dissolution of the precursor step to the supercritical drying step, including the carbonization process for carbon aerogels, are investigated from the studies reported in the literature.

Real-space modeling for complex structures based on small-angle X-ray scattering

A three-dimensional real-space model has been created for complex hierarchical materials by matching observed and simulated small-angle X-ray scattering patterns. The simulation is performed by arranging the positions of small primary particles and constructing an aggregate structure in a finite-sized cell.

A three-dimensional real-space model has been created for hierarchical materials by matching observed and simulated small-angle X-ray scattering patterns. The simulation is performed by arranging the positions of small primary particles and constructing an aggregate structure in a finite-sized cell. In order to avoid the effect of the finite size of the cell, the cell size is extended to infinity by introducing an asymptotic form of the long-range correlations among the primary particles. As a result, simulations for small-angle X-ray scattering patterns can be performed correctly in the low-wavenumber regime (<0.1 nm⁻¹), allowing the model to handle hundred-nanometre-scale structures composed of primary particles of a few nanometres in size. An aerogel structure was determined using this model, resulting in an excellent match with the experimental scattering pattern. The resultant three-dimensional model can generate cross-sectional images similar to those obtained by transmission electron microscopy, and the calculated pore-size distribution is in accord with that derived from the gas adsorption method.

Machine learning-based structure-property predictions in silica aerogels

The structural features in silica aerogels are known to be modelled effectively by the diffusion-limited cluster-cluster aggregation (DLCA) approach. In this paper, an artificial neural network (ANN) is developed for predicting the fractal properties of silica aerogels, given the input parameters for a DLCA algorithm. This approach of machine learning substitutes the necessity of first generating the DLCA structures and then simulating and characterising their fractal properties. The developed ANN demonstrates the capability of predicting the fractal dimension for any given set of DLCA parameters within an accuracy of R² = 0.973. Furthermore, the same ANN is subsequently inverted for predicting the input parameters for reconstructing a DLCA model network of silica aerogels, for a given desired target fractal dimension. There, it is shown that the fractal dimension is not a unique characteristic defining the network structure of silica aerogels, and the same fractal dimension can be obtained for different sets of DLCA input parameters. However, the problem of non-uniqueness is solved by using a guided gradient descent approach for predictive modelling purposes within certain bounds of the input parameter-space. Model DLCA structures are generated from the constrained and unconstrained inversion, and are compared against several parameters, amongst them, the pore-size distributions. The constrained inversion of the ANN is shown to predict the DLCA model parameters for a desired fractal dimension within an error of 2%.

Self-assembly of the cationic surfactant n-hexadecyl-trimethylammonium chloride in methyltrimethoxysilane aqueous solution: classical and reactive molecular dynamics simulations

A flexible aerogel polymerized from methyltrimethoxysilane (MTMS) shows great promise as a high-performance insulator owing to its substantially low thermal conductivity and mechanical flexibility, attributed to its porous microstructure and organic-inorganic hybridization, respectively, which promote its industrial applications. Conventionally, the cationic surfactant n-hexadecyltrimethylammonium chloride (CTAC) is utilized to experimentally control the nanoscale microstructure and, consequently, the flexibility of the MTMS aerogel; however, the mechanism through which CTAC prevents MTMS aggregation in the solution is not yet fully understood. This study unravels the role of CTAC in preventing MTMS aggregation in aqueous solution using both classical and reactive molecular dynamics simulations. We found that CTAC molecules can form self-aggregates even when the polymerization of MTMS progresses and then the MTMS-derived oligomer turns to be hydrophobic in aqueous solution. In summary, the self-assemblies of CTAC disperse among the MTMS associations and effectively prevent MTMS clustering, and this is considered as the key mechanism underlying the formation of a flexible microstructure of the hybrid aerogel.

Cellular Automata Modeling of Silica Aerogel Condensation Kinetics

The formation of silica aerogels and the kinetics of condensation were investigated numerically. The influence of the reaction-limited to the diffusion-limited aggregation (RLA to DLA) transition on the reaction kinetics curves and the evolution of the aggregate size distribution during condensation were examined. The 2D cellular automaton was developed and applied to reflect the process of secondary particle aggregation. Several tendencies were observed due to the adjustment of the model parameters: the probability of condensation reaction and the particles' concentration. The final wet-gel structures' visualizations proves that the structure becomes more dense and compact due to entering the RLA regime. The simulation time (associated with the gelation time) decreased along with the increase in both model parameters. The lower the collision probability, the slower reaction becomes, and particles are more likely to penetrate the structure deeper until they finally join the aggregate. The developed model reflects the condensation process's nature and its mechanisms properly and indicates a significant potential for further aerogel synthesis investigations and for the prediction of wet-gel properties according to condensation parameters.

Modeling and Simulation of the Aggregation and the Structural and Mechanical Properties of Silica Aerogels

Mechanical properties of aerogels are controlled by the connectivity of their network. In this paper, in order to study these properties, computational models of silica aerogels with different morphological entities have been generated by means of the diffusion-limited cluster-cluster aggregation (DLCA) algorithm. New insights into the influence of the model parameters on the generated aerogel structures and on the finite deformation under mechanical loads are provided. First, the structural and fractal properties of the modeled aerogels are investigated. The dependence of morphological properties such as the particle radius and density on these properties is studied. The results are correlated with experimental small-angle X-ray scattering (SAXS) data of a silica aerogel. The DLCA models of silica aerogels are analyzed for their mechanical properties with finite element simulations. There, the aerogel particles are modeled as nodes and the interparticle bonds as beam elements to account for bond stretching, bending, and torsion. The scaling relation between the elastic moduli E and relative density ρ, E ∝ ρ^m, is investigated and the exponent m = 3.61 is determined. Backbone paths evidently appear in the 3-d network structure under deformation, while the majority of the bonds in the network do not bear loads. The sensitivity of particle neck-sizes on the mechanical properties is also studied. All the results are shown to be qualitatively as well as quantitatively in agreement with the experimental data or with the available literature.

Reaction Rate Governs the Viscoelasticity and Nanostructure of Folded Protein Hydrogels

Hydrogels constructed from folded protein domains are of increasing interest as resilient and responsive biomaterials, but their optimization for applications requires time-consuming and costly molecular design. Here, we explore a complementary approach to control their properties by examining the influence of crosslinking rate on the structure and viscoelastic response of a model hydrogel constructed from photochemically crosslinked bovine serum albumin (BSA). Gelation is observed to follow a heterogeneous nucleation pathway in which BSA monomers crosslink into compact nuclei that grow into fractal percolated networks. Both the viscoelastic response probed by shear rheology and the nanostructure probed by small-angle X-ray scattering (SAXS) are shown to depend on the photochemical crosslinking reaction rate, with increased reaction rates corresponding to higher viscoelastic moduli, lower fractal dimension, and higher fractal cluster size. Reaction rate-dependent changes are shown to be consistent with a transition between diffusion- and rate-limited assembly, and the corresponding changes to viscoelastic response are proposed to arise from the presence of nonfractal depletion regions, as confirmed by SAXS. This controllable nanostructure and viscoelasticity constitute a potential route for the precise control of hydrogel properties, without the need for molecular modification.

Anomalous Anisotropic Nanoparticle Aggregation in Cu2(OH)3Br Gels

Aerogels are of interest for their ability to uniformly incorporate nanoscale features into macroscopic assemblies, which enabled applications that require low density, high surface area, and/or bicontinuous networks. The structure of the nanoporous network is intrinsically linked to the macroscopic properties of aerogels. Hence, control of this structure is of paramount importance. Small-angle X-ray scattering (SAXS) is used here to monitor nanoparticle aggregation in situ in Cu₂(OH)₃Br aerogels formed via epoxide-assisted gelation. Anomalous anisotropic aggregation is observed in the absence of templating agents and is attributed to the molecular structure of the inorganic nanoparticles themselves. This is a fundamental departure from the models currently used to describe traditional inorganic sol-gel chemistry where nanoparticles are believed to undergo isotropic diffusion- and/or kinetically limited aggregation. Time-resolved SAXS indicates that Cu₂(OH)₃Br nanoparticles nucleate rapidly from solution to form unbranched chain-like aggregates rather than branched mass-fractal aggregates. Sizes of primary particles (∼1.5 nm) and the chain-like structure of their aggregates are independent of particle concentration (gel density), while rates of particle aggregation, gelation time, and aggregate size are strongly dependent upon particle concentration, which implies that the chemistry of particle formation and the physics of particle aggregation are independent processes. Because the conditions necessary for creating anisotropic structures are not unique to Cu₂(OH)₃Br, these results could provide insight into the structure and gelation mechanisms of other inorganic aerogels.

Using Patchy Particles to Prevent Local Rearrangements in Models of Non-equilibrium Colloidal Gels

Simple models based on isotropic interparticle attractions often fail to capture experimentally observed structures of colloidal gels formed through spinodal decomposition and subsequent arrest: the resulting gels are typically denser and less branched than their experimental counterparts. Here, we simulate gels formed from soft particles with directional attractions ("patchy particles"), designed to inhibit lateral particle rearrangement after aggregation. We directly compare simulated structures with experimental colloidal gels made using soft attractive microgel particles, by employing a "skeletonization" method that reconstructs the three-dimensional backbone from experiment or simulation. We show that including directional attractions with sufficient valency leads to strongly branched structures compared to isotropic models. Furthermore, combining isotropic and directional attractions provides additional control over aggregation kinetics and gel structure. Our results show that the inhibition of lateral particle rearrangements strongly affects the gel topology and is an important effect to consider in computational models of colloidal gels.

Computer simulations of colloidal gels: how hindered particle rotation affects structure and rheology

The effects of particle roughness and short-ranged non-central forces on colloidal gels are studied using computer simulations in which particles experience a sinusoidal variation in energy as they rotate. The number of minima n and energy scale K are the key parameters; for large K and n, particle rotation is strongly hindered, but for small K and n particle rotation is nearly free. A series of systems are simulated and characterized using fractal dimensions, structure factors, coordination number distributions, bond-angle distributions and linear rheology. When particles rotate easily, clusters restructure to favor dense packings. This leads to longer gelation times and gels with strand-like morphology. The elastic moduli of such gels scale as G'∝ω^0.5 at high shear frequencies ω. In contrast, hindered particle rotation inhibits restructuring and leads to rapid gelation and diffuse morphology. Such gels are stiffer, with G'∝ω^0.35. The viscous moduli G'' in the low-barrier and high-barrier regimes scale according to exponents 0.53 and 0.5, respectively. The crossover frequency between elastic and viscous behaviors generally increases with the barrier to rotation. These findings agree qualitatively with some recent experiments on heterogeneously-surface particles and with studies of DLCA-type gels and gels of smooth spheres.

Mechanics of Nanostructured Porous Silica Aerogel Resulting from Molecular Dynamics Simulations

Silica aerogels are nanostructured, highly porous solids which have, compared to other soft materials, special mechanical properties, such as extremely low densities. In the present work, the mechanical properties of silica aerogels have been studied with molecular dynamics (MD) simulations. The aerogel model of 192 000 atoms was created with different densities by direct expansion of β-cristobalite and subjected to series of thermal treatments. Because of the high number of atoms and improved modeling procedure, the proposed model was more stable and showed significant improvement in the smoothness of the resulting stress-strain curves in comparison to previous models. Resulting Poisson's ratio values for silica aerogels lie between 0.18 and 0.21. The elasticity moduli display a power law dependence on the density, with the exponent estimated to be 3.25 ± 0.1. These results are in excellent agreement with reported experimental as well as computational values. Two different deformation scenarios have been discussed. Under tension, the low-density aerogels were more ductile while the denser ones behaved rather brittle. In the compression simulations of low-density aerogels, deformation occurred without significant increase in stress. However, for high densities, atoms offer a higher resistance to the deformation, resulting in a more stiff response and an early densification. The relationship between different mechanical parameters has been found in the cyclic loading simulations of silica aerogels with different densities. The residual strain grows linearly with the applied strain (≥0.16) and can be approximated by a phenomenological relation ϵ_p = 1.09ϵ_max - 0.12. The dissipation energy also varies with the compressive strain according to a power law with an exponent of 2.31 ± 0.07. Moreover, the tangent modulus under cyclic loading varies exponentially with the compressive strain. The results of the study pave the way toward multiscale modeling of silica as well as reinforced silica aerogels.

Micro-mechanical modelling of cellulose aerogels from molten salt hydrates

In this paper, a generalised micro-mechanical model capable of capturing the mechanical behaviour of polysaccharidic aerogels, in particular cellulose aerogels, is proposed. To this end, first the mechanical structure and properties of these highly nanoporous cellulose aerogels prepared from aqueous salt hydrate melts (calcium thiocyanate, Ca(SCN)2·6H2O and zinc chloride, ZnCl2·4H2O) are studied. The cellulose content within these aerogels is found to have a direct relation to the microstructural quantities such as the fibril length and diameter. This, along with porosity, appears to influence the resulting mechanical properties. Furthermore, experimental characterisation of cellulose aerogels was done using scanning electron microscopy (SEM), pore-size data analysis, and compression tests. Cellulose aerogels are of a characteristic cellular microstructures and accordingly a network formed by square shaped cells is considered in the micro-mechanical model proposed in this paper. This model is based on the non-linear bending and collapse of such cells of varying pore sizes. The extended Euler-Bernoulli beam theory for large deflections is used to describe the bending in the cell walls. The proposed model is physically motivated and demonstrates a good agreement with our experimental data of both ZnCl2 and Ca(SCN)2 based cellulose aerogels with different cellulose contents.

Molecular dynamics study of coagulation in silica-nanocolloid-water-NaCl systems based on the atomistic model

In the present work, large scale molecular dynamics (MD) simulations of nanocolloidal silica in aqueous NaCl solutions were performed using a fully atomistic model to study the microscopic structures and dynamics of the systems that lead to aggregation or gelation. Our attention is focused on the self-organizations that occur in the structures of the colloidal silica and water for various concentrations of NaCl. As the salt concentration increased, coagulation developed through the direct bonding of SiO4 units. The trend was explained by the systematic changes in the pair correlation functions related to the barrier height in the potential of mean force [J. G. Kirkwood, J. Chem. Phys., 1935, 3, 300]. Network structures of silica were visualised, and their fractal dimensions were examined by computing the running coordination numbers of Si-Si pairs and also by the analysis of two dimensional images. The calculated dimension by the former method was comparable to the experimental observations for the aggregation of silica colloids, and at longer length scales, super-aggregation was evident in the gelation process. The result from the 2D images is found to be insensitive to the differences in the structure. Clear changes in both the structure and mobility of the water were observed as the NaCl concentration increased, suggesting the importance of the solvent structures to these processes, although such a feature is lacking in the conventional models and most simulations of colloids.

Simulation of the tensile properties of silica aerogels: the effects of cluster structure and primary particle size

A new two-level model is proposed to investigate the relationship between the mechanical properties and microstructure of silica aerogels. This two-level model consists of the particle-particle interaction model and the cluster structure model. The particle-particle interaction model is proposed to describe interactions between primary particles, in which the polymerization reaction between primary particles is considered. The cluster structure model represents the geometrical structure of silica aerogels, and it is established using a modified diffusion-limited colloid aggregation (DLCA) algorithm. This two-level model is used to investigate the tensile behavior of silica aerogels based on the discrete element method (DEM). The numerical results show that the primary particle size has significant effects on the elastic modulus and tensile strength of silica aerogels. Moreover, the power-law relationships between tensile properties and aerogel density are dependent on the variation of the primary particle radius with density. The present results can explain the difference among different experimental exponents to a certain extent. In comparison with experimental data within a wide density range, this two-level model provides good quantitative estimations of the elastic modulus and tensile strength of silica aerogels after the size effects of the primary particle are considered. This paper provides a fundamental understanding of the relationship between the mechanical properties and microstructure of silica aerogels. The two-level model can be extended to study the mechanical properties of other aerogels and aerogel composites.

Small-angle scattering of dense, polydisperse granular porous media: computation free of size effects

Small-angle x-ray and neutrons scattering is a widespread experimental tool for the investigation of the microstructure of random heterogeneous materials. Validation of (computer-generated) model microstructures often requires the numerical computation of the scattering intensity, which must be carried out with great care due to finite size effects. In this paper, a new method for this computation is presented. It is superior to previously existing methods for three reasons: First, it applies to any type of microstructure (not necessarily granular). Second, closed-form expressions of the size effects inherent to the proposed method can be rigorously derived and removed (in this sense, our method is free of size effects). Third, the complexity of the new algorithm is linear and the computation can easily be updated to account for local changes of the microstructure, while most existing algorithms are quadratic and any change of the microstructure requires a full recomputation. The present paper provides full derivation and validation of this method. Application to the computation of the scattering intensity of dense, polydisperse assemblies of spheres is then presented. A new, simple algorithm for the generation of these dense configurations is introduced. Finally, the results are critically reviewed in the perspective of hardened cement pastes.

Small-angle x-ray scattering measurements of the microstructure of liquid helium mixtures adsorbed in aerogel

Small-angle x-ray scattering (SAXS) was used to measure the microstructure of isotopic mixtures of 3He and 4He adsorbed into silica aerogels as a function of temperature and 3He concentration. The SAXS measurements could be well described by the formation of a nearly pure film of 4He which separates from the bulk mixture onto the aerogel strands and which thickens with decreasing temperature. Previous observations of a superfluid 3He -rich phase are consistent with superfluidity existing within this film phase. Observed differences between different density aerogels are explained in terms of the depletion of 4He from the bulk mixture due to film formation.

Computer simulation of model cohesive powders: influence of assembling procedure and contact laws on low consolidation states

Molecular dynamics simulations are used to investigate the structure and mechanical properties of a simple two-dimensional model of a cohesive granular material. Intergranular forces involve elasticity, Coulomb friction, and a short-range attraction akin to the van der Waals force in powders. The effects of rolling resistance (RR) at intergranular contacts are also studied. The microstructure of the cohesive packing under low pressure is shown to depend sensitively on the assembling procedure which is applied to the initially isolated particles of a granular gas. While a direct compression produces a final equilibrated configuration with a similar density to that of cohesionless systems, the formation of large aggregates prior to the application of an external pressure results in much looser stable packings. A crucial state variable is the ratio P;{*}=PaF_{0} of applied pressure P , acting on grains of diameter a , to maximum tensile contact force F0 . At low P;{*} the force-carrying structure and force distribution are sensitive to the level of velocity fluctuations in the early stages of cluster aggregation. The coordination number of packings with RR approaches 2 in the limit of low initial velocities or large rolling friction. In general the force network is composed of hyperstatic clusters, typically comprising four to a few tens of grains, in which forces reach values of the order of F0 , joined by barely rigid arms, where contact forces are very small. Under growing P;{*} , it quickly rearranges into force chainlike patterns that are more familiar in dense systems. Density correlations are interpreted in terms of a fractal structure, up to a characteristic correlation length xi of the order of ten particle diameters for the studied solid fractions. The fractal dimension in systems with RR coincides, within measurement uncertainties, with the ballistic aggregation result, in spite of a possibly different connectivity, but is apparently higher without RR. Possible effects of micromechanical and assembling process parameters on mechanical strength of packings are evoked.

Critical scaling properties at the superfluid transition of 4He in aerogel

We study the superfluid transition of 4He in aerogel by Monte Carlo simulations and finite size scaling analysis. Aerogel is a highly porous silica glass, which we model by a diffusion limited cluster aggregation model. The superfluid is modeled by a three dimensional XY model, with excluded bonds to sites on the aerogel cluster. We obtain the correlation length exponent nu=0.73+/-0.02, in reasonable agreement with experiments and with previous simulations. For the heat capacity exponent alpha, both experiments and previous simulations suggest deviations from the Josephson hyperscaling relation alpha=2-dnu. In contrast, our Monte Carlo results support hyperscaling with alpha=-0.2+/-0.05. We suggest a reinterpretation of the experiments, which avoids scaling violations and is consistent with our simulation results.

Phase separation and percolation of reversibly aggregating spheres with a square-well attraction potential

Reversible aggregation of spheres is simulated using a novel method in which clusters of bound spheres diffuse collectively with a diffusion coefficient proportional to their radius. It is shown that the equilibrium state is the same as with other simulation techniques, but with the present method more realistic kinetics are obtained. The behavior as a function of volume fraction and interaction strength was tested for two different attraction ranges. The binodal and the percolation threshold were determined. The cluster structure and size distribution close to the percolation threshold were found to be consistent with the percolation model. Close to the binodal phase separation occurred through the growth of spherical dense domains, while for deep quenches a system spanning network is formed that coarsens with a rate that decreases with increasing attraction. We found no indication for arrest of the coarsening.

Menger sponge-like fractal body created by a novel template method

We have established experimental strategies on how to create a Menger sponge-like fractal body and how to control its fractal dimension. The essence was to utilize alkylketene dimer (AKD), which spontaneously forms super-water-repellent fractal surface. We prepared "fractal AKD particles" with fractal surface structure as templates of pores in fractal body. The fractal body was synthesized by filling the remained space between the packed template particles with a tetramethyl orthosilicate solution, solidifying it by the sol-gel process, and removing the template by calcinations. We have succeeded in systematically creating fractal bodies of silica with different cross-sectional fractal dimensions D(cs)=1.87, 1.84, and 1.80 using "fractal template particles" compressed under the ratio=1.0, 2.0, and 3.0, respectively. We also discussed the possibilities of their fractal geometries in comparison with mathematical models. We concluded that the created fractal bodies were close to a Menger sponge and its modified one. Our experimental strategy allows us to design fractality of porous materials.

Gas adsorption and desorption in silica aerogels: a theoretical study of scattering properties

We present a numerical study of the structural correlations associated with gas adsorption and desorption in silica aerogels in order to provide a theoretical interpretation of scattering experiments. Following our earlier work, we use a coarse-grained lattice-gas description and determine the nonequilibrium behavior of the adsorbed gas within a local mean-field analysis. We focus on the differences between the adsorption and desorption mechanisms and their signature in the fluid-fluid and gel-fluid structure factors as a function of temperature. At low temperature, but still in the regime where the isotherms are continuous, we find that the adsorbed fluid density, during both filling and draining, is correlated over distances that may be much larger than the gel correlation length. In particular, extended fractal correlations may occur during desorption, indicating the existence of a ramified cluster of vapor filled cavities. This also induces an important increase of the scattering intensity at small wave vectors. The similarity and differences with the scattering of fluids in other porous solids such as Vycor are discussed.

Dynamics of capillary condensation in aerogels

Dynamics of capillary condensation of liquid 4He in various density silica aerogels was investigated systematically. Interfaces were clearly visible when bulk liquid was rapidly sucked into the aerogel. Time evolution of the interface positions was consistent with the Washburn model and their effective pore radii were obtained. Condensation was a single step in a dense aerogel and two steps in a low density aerogel. Crossover between the two types of condensation was observed in an intermediate density aerogel. Variety of the dynamics may be the manifestation of the fractal nature of aerogels which had a wide range of distribution of pore radii.

Influence of the Brownian step size in off-lattice Monte Carlo simulations of irreversible particle aggregation

The influence of the Brownian step size in off-lattice Monte Carlo simulations of the aggregation and gelation of spheres is studied. It is found that the kinetics are strongly influenced if the step size is larger than the mean smallest distance between the sphere surfaces. The structure of the clusters and the gels is influenced, but only over length scales smaller than the step size. Using large step sizes leads to a narrower size distribution of the clusters. Implications of the present results are discussed for simulations reported in the literature in which the Brownian step size was chosen equal to the sphere diameter.

Influence of surface chemistry on the SAXS response of polymer-based activated carbons

Small-angle X-ray scattering (SAXS) measurements using contrast variation are reported for activated carbons prepared from poly(ethyleneterephthalate). The carbon surfaces are functionalized to different degrees by cold and hot nitric acid treatment. The latter treatment reduces the surface area by 75%, but the pore size distribution in the micropore range is hardly affected. Seven liquids, n-hexane, i-octane, i-propanol, cyclohexane, toluene, alpha-pinene, and nitrobenzene, in addition to water vapor, were used as contrast modifiers. Although the values of the specific surface area S(X) deduced from these measurements are relatively insensitive to the solvent, the detailed SAXS spectra reveal interactions occurring on different spatial scales that depend on the surface chemistry of the carbon and on the physicochemical properties of the solvent. In the most heavily oxidized sample, the amphiphilic molecule i-propanol stabilizes the surface structure, whereas nonpolar molecules make the rough surface smoother. In the untreated and the lightly functionalized carbons, water vapor at 50% relative humidity condenses only partially in the micropores at room temperature, whereas in the heavily treated sample condensation in the micropores is practically complete.

NMR measurements of hyperpolarized He3 gas diffusion in high porosity silica aerogels

Hyperpolarized 3He is used to non-destructively probe by NMR the structure of custom-made and commercial silica aerogels (97% and 98.5% porous). Large spin-echo signals are obtained at room temperature and very low magnetic field (2 mT) even with small amounts of gas. Attenuation induced by applied field gradients results from the combined effects of gas diffusion and confinement by the porous medium on atomic motion. Nitrogen is used as a buffer gas to reach equivalent 3He pressures ranging from 5 mbars to 3.5 bars. The observed pressure dependence suggests a nonuniform structure of the aerogels on length scales up to tens of micrometers. A description by broad phenomenological distributions of mean free paths is proposed, and quantitatively discussed by comparison to numerical calculations. The investigated aerogel samples exhibit different effective diffusion characteristics despite comparable nominal porosities.

Application of the Bethe-Peierls approximation to a lattice-gas model of adsorption on mesoporous materials

We calculate adsorption and desorption isotherms in models of several classes of porous materials using a lattice-gas model solved in the Bethe-Peierls (quasichemical) approximation. Isotherms and fluid density profiles from the Bethe-Peierls and Bragg-Williams approximations are compared with grand-canonical Monte Carlo simulation results. The Bethe-Peierls approximation produces both more accurate adsorption and desorption isotherms and more realistic fluid density profiles than the Bragg-Williams approximation. Details of the application of the Bethe-Peierls approximation applied to a three-dimensionally inhomogeneous system are given. We show that the numerical solution of this theory can be accomplished using a self-consistent iterator very similar to that currently used in studies employing the Bragg-Williams approximation. This iterative scheme is substantially more efficient than the numerical optimization method used in many previous studies of lattice-gas models in the quasichemical approximation. We find that use of the Bethe-Peierls approximation is only slightly more computationally demanding than the Bragg-Williams approximation, and thus recommend it for use in future work on this class of models.

Fractal dimensions of silica gels generated using reactive molecular dynamics simulations

We have used molecular dynamics simulations based on a three-body potential with charge transfer to generate nanoporous silica aerogels. Care was taken to reproduce the sol-gel condensation reaction that forms the gel backbone as realistically as possible and to thereby produce credible gel structures. The self-similarity of aerogel structures was investigated by evaluating their fractal dimension from geometric correlations. For comparison, we have also generated porous silica glasses by rupturing dense silica and computed their fractal dimension. The fractal dimension of the porous silica structures was found to be process dependent. Finally, we have determined that the effect of supercritical drying on the fractal nature of condensed silica gels is not appreciable.

Computer simulation of the phase diagram for a fluid confined in a fractal and disordered porous material

We present a grand canonical Monte Carlo simulation study of the phase diagram of a Lennard-Jones fluid adsorbed in a fractal and highly porous aerogel. The gel environment is generated from an off-lattice diffusion limited cluster-cluster aggregation process. Simulations have been performed with the multicanonical ensemble sampling technique. The biased sampling function has been obtained by histogram reweighting calculations. Comparing the confined and the bulk system liquid-vapor coexistence curves we observe a decrease of both the critical temperature and density in qualitative agreement with experiments and other Monte Carlo studies on Lennard-Jones fluids confined in random matrices of spheres. At variance with these numerical studies we do not observe upon confinement a peak on the liquid side of the coexistence curve associated with a liquid-liquid phase coexistence. In our case only a shouldering of the coexistence curve appears upon confinement. This shoulder can be associated with high density fluctuations in the liquid phase. The coexisting vapor and liquid phases in our system show a high degree of spatial disorder and inhomogeneity.

Mechanisms for gas adsorption and desorption in silica aerogels: the effect of temperature

We present a theoretical study of the adsorption and desorption mechanisms of fluids in silica aerogels, focusing on the effect of temperature. We adopt a coarse-grained lattice description in which the gel structure is generated by a diffusion-limited cluster-cluster aggregation algorithm and the fluid configurations are computed using local mean-field (i.e., density functional) theory. Our calculations reproduce qualitatively the changes in the shape of the hysteresis loops observed with (4)He in gels of varying porosity. We study in detail the morphology of the condensation and evaporation events that correspond to the irreversible processes (avalanches) which are at the origin of the hysteresis. Depending on porosity and temperature, these avalanches may be localized, involve regions that extend beyond the gel correlation length, or even span the entire sample. This makes difficult the characterization of aerogels based on analyzing sorption isotherms.

Reverse Monte Carlo modeling of the structure of colloidal aggregates

In this work we present results for the structure of aerogels coming from the diffusion-limited cluster aggregation simulation method. Pair distribution functions and structure factors, resulting from simulation, were considered as experimental input for reverse Monte Carlo modeling. The modeling yielded structural models with pair distribution functions and structure factors nearly identical to the results of the simulations. Particle configurations from both the simulations and reverse Monte Carlo modeling have been analyzed in terms of the distribution of the number of neighbors. It is suggested that the reverse Monte Carlo method, when applied to the structure factor, may be a suitable technique for the interpretation of experimental scattering data on colloidal aerogels.

Experimental investigation of colloidal gel structures

We investigate experimentally the structural properties of colloidal gels, formed under both diffusion-limited and reaction-limited aggregation conditions, using light scattering measurements and compare the results with the literature Monte Carlo (MC) simulations. The scattering structure factors have been measured for the two classes of gels in the range of the particle volume fractions between 0.02 and 0.07. From these, the corresponding fractal dimension values have been estimated. These have been found to be in good agreement with those estimated from the structure factors computed from MC simulated gels. On the basis of our previous research (Lattuada et al. Langmuir 2003, 19, 6312), this confirms that the scattering structure factor of a gel provides erroneously a small fractal dimension value, which decreases as the particle volume fraction increases. Furthermore, it is observed that the average size of the fractal clusters is larger in real gels than in simulated gels.

Morphological investigation of chemically treated poly(ethylene terephthalate)-based activated carbons

Complementary techniques, including low-temperature nitrogen adsorption and small-angle X-ray scattering (SAXS), are applied to detect the effects of surface functionalization on the morphology of activated carbon derived from poly(ethylene terephthalate) (PET). Scanning electron microscopy (SEM) is also employed as an auxiliary method to visualize the surface below the micron scale. The SEM images reveal a micron-sized ridgelike texture. Room temperature acid treatment makes the ridges become more pronounced, while treatment with boiling acid uncovers fiberlike structures of roughly 1 microm diameter. All samples display an apparent surface fractal dimension of Ds = 2.4 in the wave vector range 0.001-0.02 A(-1). Nitric acid at room temperature increases the surface oxygen content only by 3 at. %, while all the adsorption properties and structural parameters reported in this paper are virtually unaffected. Significant differences in the morphology at submicron scales appear only after boiling acid treatment. The resulting carbon remains highly microporous, but the loss of Brunauer-Emmett-Teller (BET) surface area from about 1150 to 304 m2/g is approximately 75%. In addition to the principal peak at around 8 A, fresh peaks appear in the polydisperse Horvath-Kawazoe (HK) pore size distribution owing to the burnoff of intervening walls. The average width of the slitlike pores calculated from the Dubinin-Radushkevich (DR) plot increases from 8.4 to 11 A. The minimum slit width where the applied probe molecules, that is, nitrogen and hexane, can enter increases from about 5 to about 5.4 A. The separation distance of the basic structural units is practically unchanged. When, however, this carbon is in contact with hexane, this distance expands from about 19 to 27 A. The swelling is consistent with the deformable nature of this sample also illustrated by the low-pressure hysteresis and the reduced helium density. Particular attention was paid to the surface areas derived from low-temperature nitrogen adsorption and X-ray measurements. Owing to the wide spatial range of the structures in these samples, estimates of the specific surface area of activated carbons can be substantially in error unless both upper and lower q ranges of the SAXS spectra are taken into account. Surface areas derived from the adsorption data either by the BET or the DR approaches were always below the values obtained by standard SAXS. As an example, the carbon sample functionalized at room temperature gave surface area values of 1114, 1293, and 1970 m2/g, respectively. The possibility that this difference is caused by inaccessible pores was excluded by contrast variation measurements with hexane.

Local mean-field study of capillary condensation in silica aerogels

We apply local mean-field (i.e., density functional) theory to a lattice model of a fluid in contact with a dilute, disordered gel network. The gel structure is described by a diffusion-limited cluster aggregation model. We focus on the influence of porosity on both the hysteretic and the equilibrium behavior of the fluid as one varies the chemical potential at low temperature. We show that the shape of the hysteresis loop changes from smooth to rectangular as the porosity increases and that this change is associated with disorder-induced out-of-equilibrium phase transitions that differ in adsorption and in desorption. Our results provide insight in the behavior of 4He in silica aerogels.

New universality class for the three-dimensional XY model with correlated impurities: application to 4He in aerogels

Encouraged by experiments on 4He in aerogels, we confine planar spins in the pores of simulated aerogels (diffusion limited cluster-cluster aggregation) in order to study the effect of quenched disorder on the critical behavior of the three-dimensional XY model. Monte Carlo simulations and finite-size scaling are used to determine critical couplings K(c) and exponents. In agreement with experiments, clear evidence of change in the thermal critical exponents nu and alpha is found at nonzero volume fractions of impurities. These changes are explained in terms of hidden long-range correlations within disorder distributions.

Dangling bond deflection model: growth of gel network with loop structure

It has been shown that the closed-loop structure in the model gel networks is responsible for their stiffness. However, the creation of loops has been underestimated in most of the existing kinetic aggregation models [e.g., DLCA (diffusion-limited cluster-cluster aggregation) and derivatives]. A dangling bond deflection (DEF) mechanism is proposed to model the fluctuation of dangling branches or dead ends under thermal excitation. The random deflections of the dangling branches can create loops in the network by forming intracluster bonds, and proceed during both the gelling and aging processes. The resulting DLCADEF networks have extensive loop structure with a negligible number of dangling branches. Its growth kinetics and fractal behavior resemble those of real gels, including volume-invariant gel time and fractal dimension of about 2. The DLCADEF model is the first attempt to model the gel growth with loop formation by the physically realistic fluctuation mechanism. The mechanical properties of the resulting networks will be studied and verified by comparison with real gels.

Colloidal aggregation with mobile impurities

The structure and aggregation kinetics of diffusion-limited cluster-cluster three-dimensional monomeric aggregates and gels is investigated as a function of the molar fraction of two types of impurities. In one case the impurities are allowed to aggregate among themselves whereas in the other the impurities are mobile monomers that remain as such during the whole aggregation process. Computer simulations are performed on a simple cubic lattice for which the functionality of the aggregating particles is effectively 6. The first type of impurity shows a decrease in the fractal dimension when compared to that of a one component system at the same concentration. As a consequence of this decrease, the gelation concentration is lowered. At higher concentrations a gelling to nongelling transition was observed. In the nongelling regime the colloidal aggregates are kept apart by the impurity clusters, developing a local ordering. For the monomeric type of impurity, at large impurity molar fractions, a nonstructured nongelling phase appears at high enough concentration, in which the colloidal aggregates are kept apart by the sea of mobile impurities that inhibits the formation of a gel. Smaller molar fractions of mobile monomeric impurities strongly affect both the fractal dimension and the kinetics of the aggregating colloid.

Scaling of the superfluid fraction and T(c) of 3He in aerogel

We have investigated the superfluid transition of 3He in different samples of silica aerogel. By comparing new measurements on a 99.5% sample with previous observations on the behavior of 3He in 98% porous aerogel, we have found evidence for a scaling of the transition temperature and superfluid density of 3He to the correlation length of the aerogel.