Dispersive shock waves in colloids with temperature dependent compressibility

Abstract

The formation of a dispersive shock wave in a colloidal medium, due to an initial jump in the light intensity, is studied. The compressibility of the colloidal particles is modeled using a series in the particle density, or packing fraction, where the virial coefficients depend on the particle interaction model. Both the theoretical hard disk and sphere repulsive models, and a model with temperature dependent compressibility, are considered. Experimental results for the second virial coefficient show that it is temperature dependent and that the particle interactions can be either repulsive or attractive. These effects are modeled using a power-law relationship. The governing equation is a focusing nonlinear Schrödinger-type equation with an implicit nonlinearity. The initial jump is resolved via a dispersive shock wave which forms before the onset of modulational instability. A semi-analytical solution is developed for the one-dimensional line bore case which predicts the amplitude of the solitary waves which form in the dispersive shock wave. The solitary wave amplitude versus jump height diagrams can exhibit three different kinds of behaviors. A unique solution, an Sshaped solution curve and multiple solution branches where the upper branch has separated from the lower branches. A bifurcation from the low to the high power branch can occur for many parameter choices as the amplitude of the initial jump increases. The effect of temperature on the evolution of the bore, the amplitude of the solitary waves and the bifurcation patterns are all discussed and the semi-analytical solutions are found to be very accurate. AbstractThe formation of a dispersive shock wave in a colloidal medium, due to an initial jump in the light intensity, is studied. The compressibility of the colloidal particles is modelled using a series in the particle density, or packing fraction, where the virial coefficients depend on the particle interaction model. Both the theoretical hard disk and sphere repulsive models, and a model with temperature dependent compressibility, are considered. Experimental results for the second virial coefficient show that it is temperature dependent and that the particle interactions can be either repulsive or attractive; these effects are modelled using a power law relationship. The governing equation is a focusing nonlinear Schrödinger-type equation with an implicit nonlinearity. The initial jump is resolved via a dispersive shock wave which forms before the onset of modulational instability. A semi-analytical solution is developed for the one dimensional line bore case which predicts the amplitude of the solitary waves which form in the dispersive shock wave. The solitary wave amplitude versus jump height diagrams can exhibit three different kinds of behaviours; an unique solution, an S-shaped solution curve and multiple solution branches where the upper branch has separated from the lower branches. A bifurcation from the low to the high power branch, can occur for many parameter choices, as the ...

Circular dispersive shock waves in colloidal media

“…Shown are the cases for k = 0 and k = 1. When the circular DSW is stationary, the result obtained is qualitatively similar to the line DSW, see Figure 7(a) [10]. As for the line DSW case, the individual waves do not completely separate, so they continue to interact with each other and are not ordered by amplitude.…”

“…However, for NLStype equations, the solitary waves amplitude is independent of wavenumber, so (9) applies for all wavenumbers [4]. We now use the semi-analytical expression for a single colloidal solitary wave [10] u(x, z) = a sech…”

“…Three different types of solitary wave amplitude versus jump height diagrams were found, depending on the background packing fraction. An extension considered a temperature dependent series form for the colloidal particle interactions [10]. Results for the line (1+1)-D DSW, using both a hard disk (HD), valid in one-dimension, and hard sphere (HS), valid in two-dimensions, theoretical models, were examined.…”

Optical dispersive shock waves in defocusing colloidal media

“…This highlights that the full constitutive law (2) is not always necessary and that Taylor expansions of it for small packing fractions can lead to adequate approximations to the full constitutive law. The constitutive law (2) constitutive relation g(η) [42]. Hopefully, this theoretical and numerical study will motivate some future experimental studies of dark DSWs in colloidal and other optical media.…”

“…While a colloid is normally a focusing medium, so that its refractive index increases with beam intensity, it can be made to be a defocusing medium [40,41], which then supports a DSW consisting of dark solitary waves at the trailing edge and linear waves at the leading edge [4, 29,42]. The DSW 65 is generated by a jump initial condition in optical beam intensity.…”