Initial stage of spinodal decomposition in a rigid-rod system

Abstract

The initial stage of spinodal decomposition is investigated for a rigid-rod system. Spinodal decomposition proceeds through either of two mechanisms: (1) The randomly aligned rods rotate toward a common director with no inherent length scale. (2) The rods diffuse axially and segregate into regions of common alignment with a selected length scale [script-l]. Previous studies on spinodal decomposition yielded radically different conclusions about which mechanism is dominant. A computational method is employed to analyze the growth rate and properties of the dominant fluctuation mode through an eigenvalue analysis of the linearized Doi diffusion equation in Fourier space. The linearized operator is discretized in Fourier mode and orientation space (k,theta,phi) space, and the maximum eigenvalue and corresponding eigenvector of the operator are calculated. Our analysis generalizes the results of previous studies and shows that the conflicts seen in those studies are due to differences in the diffusivities for rotational motion, motion perpendicular to the rod axis, and motion along the rod axis. For a given system density, a plot of the dominant perturbation wave number k(max) as a function of the diffusivity ratios shows two separate regions corresponding to mechanisms (1) and (2). High rotational diffusivity corresponds to mechanism (1), whereas high axial diffusivity corresponds to mechanism (2). The transition between the two mechanisms depends on the ratio of diffusivities and on system density. Also, the dominant wave number increases with increasing density indicating that a deeper quench into the spinodal regime leads to a smaller characteristic length scale.

Spinodal decomposition and nematic coarsening in a rigid-rod solution

“…Variations in f are restricted to the y-dimension as in previous studies of spinodal decomposition [5,6,8,9]. The system has length b and periodic boundary conditions in the y-dimension.…”

“…Each spinodal decomposition simulation begins with an unstable isotropic system of period b at concentration N; the system is perturbed by a periodic fluctuation with twist, bend, and splay characteristics corresponding to the dominant fluctuations computed through linear stability analysis [9]. Gradients in S zz − S xx and in S xz are classified as examples of the "twist" mode.…”

“…Their results also indicated that the dominant wavelength of spinodal decomposition changes over time. In a previous paper [9], we showed that these different assumptions about entanglement effects result in radically different mechanisms for spinodal decomposition. One mechanism is dominated by rod rotation with no inherent length scale, whereas the other is dominated by rod translation into misaligned ordered regions of a particular size.…”

“…Subsequent studies of the initial stage of spinodal decomposition also restricted gradients in f to one dimension [6][7][8][9]. Dhont and Briels [8] argued that the diffusivities should not be changed to reflect restricted motion effects because the interaction potential already accounts for the "caging" presence of surrounding rods.…”

Rheological phase diagrams for nonhomogeneous flows of rodlike liquid crystalline polymers

“…However, until recently, the nonhomogeneous formulation of the Doi diffusion equation [16] was used only in its linearized form for linear stability analysis of spinodal decomposition [17][18][19], because the equation is numerically unwieldy. The nonhomogeneous theory presents two major numerical challenges.…”