This paper presents a review of scattering and absorption of light by fractal aggregates. The aggregates are typically diffusion limited cluster aggregates (DLCA) with fractal dimensions of D

https://www.phys.ksu.edu/personal/sor/publications/2001/light.pdf

Light Scattering by Fractal Aggregates: A Review

Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the 'ant in the labyrinth') on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures.

Diffusion in disordered media

The interactions between charged colloidal particles with sufficient strength to cause crystallization are shown to be describable in terms of the usual Debye–Huckel approximation, but with a renormalized charge. The effective charge in general is smaller than the actual charge. We calculate the relationship between the actual charge and the renormalized charge by solving the Boltzmann–Poisson equation numerically in a spherical Wigner–Seitz cell. We then relate the numerical solutions and the effective charge to the osmotic pressure and the bulk modulus of the crystal. Our calculations also reveal that the renormalization of the added electrolyte concentration is negligible, so that the effective charge computations are useful even in the presence of salts.

Charge renormalization, osmotic pressure, and bulk modulus of colloidal crystals: Theory

A general theory on the electrostatic interaction in macroionic solution is developed. It is assumed that the motion of macroions is adiabatically cut off from that of simple ions and the distribution of simple ions is determined by the Boltzmann distribution. A generalized Poisson–Boltzmann equation is solved with the linearization approximation. The total electrostatic energy of the solution is obtained, from which the Helmholtz free energy of interaction is calculated. The Gibbs free energy is derived by the use of its additivity relation with respect to the number of the ionic species. The two free energies are demonstrated not to be equal to each other for highly charged macroions, in contrast with widely accepted view. It is shown that the electrostatic potential for a pair of macroions predicts strong attraction between the macroions, as was the case with the Levine–Dube treatment, and the Helmholtz pair potential results in purely repulsive interaction, in accordance with the DLVO theory. However, the Gibbs pair potential leads us to repulsion at small interparticle separations and attraction at large distances, creating a ‘‘secondary’’ minimum (with a potential valley deeper than the thermal energy for spherical colloidal particles). Without taking refuge in a van der Waals attraction the theory evidently substantiates the experimental fact recently reported, namely the existence of Coulombic intermacroion attraction through the intermediary of counterions. Thus, the ordering of macroions (synthetic and biological) and charged latex particles in solutions can be accounted for in terms of the present theory: By taking the lattice sum of the pair potential, it is shown that the bcc structure for polymer latex particles is favored in the small κa region (κ−1: the Debye radius, a: the radius of particle), whereas close‐packed structures dominate in the intermediate and large κa regions. In the whole regions, the simple cubic structure is disfavored. The interparticle distance (the position of the secondary minimum) is shown to become smaller with increasing κa, while the potential valley takes the deepest value at κa≊1. The interparticle distance is found to decrease with latex concentration and temperature. These theoretical outcomes are compared with experimental data currently available.

On the electrostatic interaction in macroionic solutions

A generalized hydrodynamic theory is developed for systems of interacting Brownian particles on the basis of a Fokker-Planck equation General results are derived for correlation functions, frequency- and wave-vector dependent transport coefficients Explicit expressions for moments, cumulants and the hydrodynamic limits of the transport coefficients are given For the special cases of overdamped systems with and without hydrodynamic interaction the general results are simplified As an example for the application of this approach the system of charged spherical polystyrene spheres in aqueous solution is treated in detail The generalized transport functions are evaluated in mode-mode coupling approximation and detailed numerical results are presented for various collective and single-particle properties Finally, the relationship to a corresponding Smoluchowski approach is discussed

Generalized hydrodynamics of systems of Brownian particles

The rigidity G and the viscosity ? of monodisperse polystyrene latexes were measured by the torsional quartz crystal method at 70 kHz, and an abrupt change in G and ? was found at the order-disorder transition of the latexes. The torsional quartz crystal method was improved for measurements with aqueous conductive samples, and a careful calibration showed that G and ? are obtained if the dielectric constant of the sample is known. In the ordered state of latex with a volume concentration of 9.3%, the apparent rigidity G and the viscosity ? were about 1.5?103 dyne/cm2 and 0.6 cp, respectively, while in the disordered state G was small and ? was 0.95 cp. Ordered latex is a ?crystal? formed by the screened Coulomb repulsion and is a good model not only for crystals but also for ordered structures in some biological systems.

Studies of Ordered Monodisperse Polystyrene Latexes. I. Shear Ultrasonic Measurements

Starting from a N‐particle diffusion equation, the effective particle (self‐) and concentration diffusion coefficients of interacting Brownian particles are studied theoretically. These two diffusion coefficients defined by distinct ways are different for interacting systems. By taking into account both potential and hydrodynamic interactions, the expression for the frequency‐dependent particle diffusion coefficient exact to linear order in number density is derived. Detailed calculations of both diffusion coefficients are performed for model systems of hard spheres with a repulsive or an attractive long‐range potential. The effects of potential and hydrodynamic interactions on these two diffusion coefficients are clarified. The velocity autocorrelation function is also calculated and non‐Markovian behaviors are discussed.

Diffusion coefficients of interacting Brownian particles

The viscoelastic behaviour at 40 kHz and the steady flow properties of monodisperse polystyrene latexes, i.e. aqueous suspensions of charged polystyrene spheres, were measured through an order-disorder transition as varying the salt concentration in the solvent and the volume fraction of spheres. A latex in the ordered lattice structure showed the rigidity of about 103 dyn/cm2 and the yield stress of about 1 dyn/cm2, which indicates that the ordered latex is a crystal in view of the mechanical properties. The ratio, 10-3, of the yield stress to the rigidity suggests a dislocation mechanism of the plastic flow of the ordered latex.

Studies of Ordered Monodisperse Latexes. III. Viscoelasticity and Flow Properties

The electrostatic potential around a particle in an ordered latex under a deformation is numerically calculated form the nonlinear Poisson-Boltzmann equation by an eccentric cell model. The static rigidity of an ordered latex and the order-disorder Lindemann's law of the crystal melting are obtained from the calculated potential. The calculated phase diagram agrees with the experimental phase diagram of Hachisu et al. The Debye-Huckel approximation turns out to be invalid at low ionic concentrations. On the basis of a two-continua model and the calculated static rigidity, propagation of shear waves in an ordered latex is studied and the existence of two kinds of modes is shown. The dynamical complex rigidity is calculated from the mechanical impedance at an oscillating plate. The results agree with the experiment reported in a previous paper.

Toshiya Ohtsuki

Papers

Studies of Ordered Monodisperse Polystyrene Latexes. I. Shear Ultrasonic Measurements

Diffusion coefficients of interacting Brownian particles

Studies of Ordered Monodisperse Latexes. III. Viscoelasticity and Flow Properties

Studies of Ordered Monodisperse Latexes. II. Theory of Mechanical Properties

Dynamic properties of concentrated suspensions of charged polystyrene spheres.