One-Dimensional Ferronematics in a Channel: Order Reconstruction, Bifurcations, and Multistability

Abstract

We study a model system with nematic and magnetic order, within a channel geometry modeled by an interval, [−D, D]. The system is characterized by a tensor-valued nematic order parameter Q and a vector-valued magnetization M, and the observable states are modeled as stable critical points of an appropriately defined free energy which includes a nemato-magnetic coupling term, characterized by a parameter c. We (i) derive L ∞ bounds for Q and M; (ii) prove a uniqueness result in specified parameter regimes; (iii) analyze order reconstruction solutions, possessing domain walls, and their stabilities as a function of D and c and; (iv) perform numerical studies that elucidate the interplay of c and D for multistability.

A Multi-Faceted Study of Nematic Order Reconstruction in Microfluidic Channels

“…Focusing on ω = 1 4 and replicating arguments from [13], we deduce that OR solutions, interpreted as minimizers of (1.6), converge in L 1 ([−1, 1]), almost everywhere to a map of the form…”

“…Additionally, OR solutions are not limited to purely nematic systems e.g. OR solutions exist in ferronematic systems comprising magnetic nanoparticles in NLC media [13]. Generalized OR solutions or OR-type solutions/instabilities (defined in section 4) are also observed in smectics and cholesterics.…”

Parameter dependent finite element analysis for ferronematics solutions

“…The ℓ → 0 limit is relevant for macroscopic domains or large domains, that are much larger than characteristic materialdependent and temperature-dependent nematic and magnetic correlation lengths [3]. The proof is done in several stages: analysis of the minimizers of the bulk potential reproduced from [10], a strong convergence result for global minimizers followed by convergence results for the bulk potential that largely follow from [2], followed by a delicate Bochner inequality for the energy density that combines ideas from [2] and [16]. Once we have the Bochner inequality, the ℓ-independent 𝐻 2 -bound for global energy minimizers in the ℓ → 0 limit is relatively standard, from estalished techniques in the Ginzburg-Landau theory for superconductivity although additional technical difficulties are encountered due to the four degrees of freedom in the problem.…”

“…For ferronematic systems with 𝑐 ≠ 0, the volume of work is limited. In [10], the authors analyze a dilute ferronematic suspension in a one-dimensional channel geometry with Dirichlet boundary conditions for both Q and M. The authors derive some key analytic ingredients -existence theorems, uniqueness theorems, maximum principle arguments and symmetric solution profiles. They also compute bifurcation diagrams for the solution branches as a function of ℓ and 𝑐.…”

Two-dimensional Ferronematics, Canonical Harmonic Maps and Minimal Connections

“…The Q-profile has a discrete set of non-orientable nematic defects and the M-profile exhibits line defects connecting these nematic defects, in the ε → 0 limit. Whilst the practical relevance of such studies remains uncertain, it is clear that strong theoretical underpinnings are much needed for systematic scientific progress in this field, and our work is a first powerful step in an exhaustive study of ferronematic solution landscapes [46] (also see recent work in [23], [39]).…”