The Application of Fractals to Colloidal Aggregation

Recent work on understanding colloidal aggregation phenomena using fractal geometry is reviewed. After describing a typical aggregation experiment, the concepts of fractal and fractal dimension are introduced and some simple models fit to describe the mechanism of diffusion-limited aggregation leading to fractal aggregates are presented. The old Smoluchowski’s theory for describing the kinetics of colloidal aggregation is revised using fractal concepts

A Numerical Study of Phase Transitions Inside the Pores of Aerogels

Phase transitions inside the pores of an aerogel are investigated by modelizing the aerogel structure by diffusion-limited cluster-cluster aggregation on a cubic lattice in a finite box and considering $q$-states Potts variables on the empty sites interacting via nearest-neighbours. Using a finite size scaling analysing of Monte-Carlo numerical results, it is concluded that for $q=4$ the transition changes from first order to second order as the aerogel concentration (density) increases. Comparison is made with the case $q=3$ (where the first order transition is weaker in three dimensions) and with the case $q=4$ but for randomly (non correlated) occupied sites. Possible applications to experiments are discussed.Comment: RevTex, 12 pages + 10 postscript figures compressed using "uufiles", To appear in J. of Non-Cryst. Solid

The Sol-Gel Process Simulated by Cluster-Cluster Aggregation

The pair-correlation function $g(r,t)$ and its Fourier transform, the structure factor $S(q,t)$, are computed during the gelation process of identical spherical particles using the diffusion-limited cluster-cluster aggregation model in a box. This numerical analysis shows that the time evolution of the characteristic cluster size $\xi$ exhibits a crossover close to the gel time $t_g$ which depends on the volumic fraction $c$. In this model $t_g$ tends to infinity when the box size $L$ tends to infinity. For systems of finite size, it is shown numerically that, when $t<t_g$, the wave vector $q_m$, at which $S(q,t)$ has a maximum, decreases as $S(q_m,t)^{-1/D}$, where $D$ is an apparent fractal dimension of clusters, as measured from the slo pe of $S(q,t)$ . The time evolution of the mean number of particles per cluster $\bar {n}$ is also investigated. Our numerical results are in qualitative agreement with small angle scattering experiments in several systems.Comment: RevTex, 13 pages + 9 postscript figures appended using "uufiles". To appear in J. of Non-Cryst. Solid

Small Angle Neutron Scattering of Aerogels: Simulations and Experiments

A numerical simulation of silica aerogels is performed using diffusion-limited cluster-cluster aggregation of spheres inside a cubic box (with periodic boundary conditions). The volume fraction $c$ is taken to be sufficiently large to get a gel structure at the end of the process. In the case of monodisperse spheres, the wavevector dependent scattered intensity $I(q)$ is calculated from the product of the form factor $P(q)$ of a sphere by the structure factor $S(q)$, which is related to the Fourier transform of $g(r)-1$, where $g(r)$ is the pair correlation function between sphere centers. The structure factor $S(q)$ exhibits large-$q$ damped oscillations characteristics of the short range (intra-aggregate) correlations between spheres. These oscillations influence the $I(q)$ curve in the $q$-region between the fractal regime and the Porod regime and quantitative comparisons are made with experiments on colloidal aerogels. Moreover, at small-$q$ values, $S(q)$ goes through a maximum characteristic of large range (inter-aggregate) correlations. Quantitative fits of the maximum in the experimental $I(q)$ curves of base-catalyzed aerogel are presented. In the case of polydisperse spheres, $I(q)$ is calculated directly from a single aggregate simulation. It is shown that increasing polydispersity shifts the location of the cross-over between the fractal and Porod regimes towards low $q$-value.Comment: RevTex, 9 pages + 11 postscript figures, compressed using "uufiles". Proceeding of the 4th International Simposium on Aerogels (To appear in J. of Non-Cryst. Solids

Scaling Theory and Numerical Simulations of Aerogel Sintering

A simple scaling theory for the sintering of fractal aerogels is presented. The densification at small scales is described by an increase of the lower cut-off length $a$ accompanied by a decrease of the upper cut-off length $\xi$, in order to conserve the total mass of the system. Scaling laws are derived which predict how $a$, $\xi$ and the specific pore surface area $\Sigma$ should depend on the density $\rho$. Following the general ideas of the theory, numerical simulations of sintering are proposed starting from computer simulations of aerogel structure based on a diffusion-limited cluster-cluster aggregation gelling process. The numerical results for $a$, $\xi$ and $\Sigma$ as a function of $\rho$ are discussed according to the initial aerogel density. The scaling theory is only fully recovered in the limit of very low density where the original values of $a$ and $\xi$ are well separated. These numerical results are compared with experiments on partially densified aerogels.Comment: RevTex, 17 pages + 6 postscript figures appended using "uufiles". To appear in J. of Non-Cryst. Solid

Small Angle Scattering by Fractal Aggregates: A Numerical Investigation of the Crossover Between the Fractal Regime and the Porod Regime

Fractal aggregates are built on a computer using off-lattice cluster-cluster aggregation models. The aggregates are made of spherical particles of different sizes distributed according to a Gaussian-like distribution characterised by a mean $a_0$ and a standard deviation $\sigma$. The wave vector dependent scattered intensity $I(q)$ is computed in order to study the influence of the particle polydispersity on the crossover between the fractal regime and the Porod regime. It is shown that, given $a_0$, the location $q_c$ of the crossover decreases as $\sigma$ increases. The dependence of $q_c$ on $\sigma$ can be understood from the evolution of the shape of the center-to-center interparticle-distance distribution function.Comment: RevTex, 4 pages + 6 postscript figures, compressed using "uufiles", published in Phys. Rev. B 50, 1305 (1994

Fluctuating Bond Aggregation: a Model for Chemical Gel Formation

The Diffusion-Limited Cluster-Cluster Aggregation (DLCA) model is modified by including cluster deformations using the {\it bond fluctuation} algorithm. From 3$d$ computer simulations, it is shown that, below a given threshold value $c_g$ of the volumic fraction $c$, the realization of all intra-aggregate bonding possibilities prevents the formation of a gelling network. For $c>c_g$, the sol-gel transition occurs at a time $t_g$ which, in contrast to DLCA, doesnot diverge with the box size. Several results are reported including small angle scattering curves and possible applications are discussed.Comment: RevTex, 9 pages + 3 postscript figures appended using "uufiles". To appear in Phys. Rev. Let

Channel diffusion of sodium in a silicate glass

We use classical molecular dynamics simulations to study the dynamics of sodium atoms in amorphous Na$_2$O-4SiO$_2$. We find that the sodium trajectories form a well connected network of pockets and channels. Inside these channels the motion of the atoms is not cooperative but rather given by independent thermally activated hops of individual atoms between the pockets. By determining the probability that an atom returns to a given starting site, we show that such events are not important for the dynamics of this system.Comment: 10 pages of Latex, 5 figures, one figure added, text expande

Computer investigation of the energy landscape of amorphous silica

The multidimensional topography of the collective potential energy function of a so-called strong glass former (silica) is analyzed by means of classical molecular dynamics calculations. Features qualitatively similar to those of fragile glasses are recovered at high temperatures : in particular an intrinsic characteristic temperature $T_c\simeq 3500$K is evidenced above which the system starts to investigate non-harmonic potential energy basins. It is shown that the anharmonicities are essentially characterized by a roughness appearing in the potential energy valleys explored by the system for temperatures above $T_c$.Comment: 5 pages; accepted for publication in PR

Anomalous Low-Field Classical Magnetoresistance in Two Dimensions

The magnetoresistance of classical two-dimensional electrons scattered by randomly distributed impurities is investigated by numerical simulation. At low magnetic fields, we find for the first time a negative magnetoresistance proportional to |B|. This unexpected behavior is shown to be due to a memory effect specific for backscattering events, which was not considered previously.Comment: 4 pages, 4 figure