## Year

2014

## Degree Name

Doctor of Philosophy

## Department

School of Mathematics and Applied Sciences

## Recommended Citation

Azmi, Amirah, Solitary waves in colloidal media, Doctor of Philosophy thesis, School of Mathematics and Applied Sciences, University of Wollongong, 2014. http://ro.uow.edu.au/theses/4355

## Abstract

Spatial solitary waves in colloidal suspensions of spherical dielectric nanoparticles are considered. If a laser light beam passes through a colloidal suspension, which consists of spherical dielectric nanoparticles, then the light beam will attract the nanoparticles, increasing the refractive index and creating an optical spatial soliton. Both the one-dimensional and two-dimensional solitary waves are considered with the hard disk and hard sphere theoretical models discussed together with results for a temperature dependent model. The interaction between the colloidal particles in the classical hard disk and sphere models are repulsive and for the temperature dependent model, the interaction between the particles can represent repulsive or attractive interactions. The interaction, or compressibility, of the colloidal particles, is modelled using a series in the particle density, or packing fraction, where the virial (or series) coefficients depend on the type of interaction model. Experimental results show that particle interactions can be temperature dependent and repulsive or attractive in nature, so we model the second virial coefficient using a physically realistic temperature power law.

Semi-analytical solitary waves, for one-dimensional and two-dimensional cases, are derived using an averaged Lagrangian and suitable trial functions for the solitary waves. Power versus propagation constant curves and neutral stability curves are obtained for both cases, which illustrate that multiple solution branches occur for the one-dimensional and two-dimensional cases. For the one-dimensional case, it is found that three solution branches (with a bistable regime) occur, while for the two-dimensional case, two solution branches (with a single stable branch) occur in the limit of low background packing fractions. The temperature dependent properties result in changes to the stability of the solitary waves, which are fully explored.

We also consider the diffraction of an optical beam in a colloidal media, for which an initial jump, or discontinuity, is resolved into a dispersive shock wave. The onedimensional semi-analytical colloidal solitary wave solutions are used together with con- servation laws to obtain a semi-analytical description of the amplitude of waves formed at the shock. When the background packing fraction is low, multiple solution branches occur for amplitude versus shock height response curve. Three solutions branches occur, with the upper stable branch detached from the unstable middle branch. At moderate background packing fraction values, an S-shaped response curve exists with all branches occurring for physically realistic parameters. When the background packing fraction is high, only a single stable solution branch occurs. This means that for low and and moderate background packing fractions, the solutions can bifurcate to the high amplitude branch, as the shock height increases.

The hard disk, hard sphere and temperature dependent models are used to describe the one-dimensional case (a line DSW), while for the two-dimensional case (a circular DSW), the hard sphere and temperature dependent models are considered. For the two-dimensional case (circular DSW) at large radius, the one-dimensional analytical results, together with geometrical considerations, provides useful semi-analytical predictions. The semi-analytical predictions and the numerical solutions are found to be in close agreement for both one-dimensional and two-dimensional dispersive shock waves.

## FoR codes (2008)

0102 APPLIED MATHEMATICS, 0105 MATHEMATICAL PHYSICS

**Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.**