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Parity Breaking in Ferrofluids with VorticityMagnetization Coupling
by Dylan Reynolds, Gustavo M. Monteiro, Sriram Ganeshan
Submission summary
Authors (as registered SciPost users):  Dylan Reynolds 
Submission information  

Preprint Link:  scipost_202309_00022v2 (pdf) 
Date submitted:  20240718 16:42 
Submitted by:  Reynolds, Dylan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
Ferrofluids are synthetic magnetic colloids consisting of magnetized nanoparticles surrounded by a repulsive surfactant layer. When subjected to an external magnetic field the ferrofluid acquires a macroscopic magnetization density which leads to magnetic behavior that is intricately coupled to the ambient fluid dynamics. Ferrofluids share several features with the chiral active fluids composed of unidirectionally spinning hematite cubes, which have been shown to possess a 2D nondissipative odd viscosity term [1]. In standard ferrofluid dynamics, 3D versions of paritybreaking terms are not commonly observed, partly because of the small size of the magnetic particles. In this work, we investigate if there are unique mechanisms in ferrofluids that can lead to a 3D odd viscosity term. Our results show that coupling the fluid vorticity ($\vec{\omega}$) to the magnetization ($\vec{M}$) with a term proportional to $\vec{\omega}\cdot\vec{M}$ leads to parity breaking in ferrofluid hydrodynamics, and results in a threedimensional odd viscosity term when the magnetization is relaxed along the direction of a uniform and static applied field. HeleShaw cells are commonly used devices to investigate ferrofluids and we demonstrate that this coupling reproduces the parity odd generalization of Darcy's Law discussed in a recent work [2]. A potential experimental setup is discussed which may reveal the presence of this coupling in a ferrofluid confined to a HeleShaw cell.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Author comments upon resubmission
We are resubmitting our manuscript entitled "Parity Breaking in Ferrofluids with VorticityMagnetization Coupling" for your consideration for publication in SciPost. We sincerely thank the referees for their time and effort in reviewing this paper. Referee 1 provided a positive review and requested additional discussion on the microscopic origins of the new paritybreaking term. In response, we have expanded the discussion to include more microscopic arguments. We have also put this coupling into context by comparing it with that of Ref. [24].
Referees 2 and 3 raised concerns about the novelty of this work and its comparisons with Ref. [24] by Markovich and Lubensky, as well as our previous work on HeleShaw cells and parity breaking fluids. We believe there is some misunderstanding, which we have addressed in the referee reports and reiterate below. Ferrofluids possess many features that break parity at the microscopic scale, but it is puzzling why, at the macroscopic scale, most observables seem to show no sign of parity breaking. To the best of our knowledge despite vast literature on the ferrofluids, there has been no discussion on the parity breaking in them. Motivated by this puzzle, our main question in this manuscript is: what are the conditions for a ferrofluid to exhibit parity breaking phenomena in a HeleShaw experiment? Our past work on HeleShaw flows motivated this work and raised whether Ferrofluids can break parity. One of the main reasons we highlighted Ref. [24] in our work is to demonstrate that following this work does not lead to any parity breaking in ferrofluids due to the small size of the magnetic particles. This is one of the results presented in our manuscript. Ref. [24] is more applicable to ferrofluids with bigger colloidal particles like those used in chiral active fluids (Ref. [1]). The novelty of our work is the new mechanism that can break parity in ferrofluids by coupling magnetization to ambient vorticity. In the end, we provide a simple experimental test for this prediction that can confirm or refute our predictions.
Finally, based on the points above, we strongly believe that our work should be considered for SciPost Physics, but in the interest of the time already spent, we would accept publishing it in SciPost CORE, if the editor still believes this to be the best fit for our current work. We will abide by the editorial decision on this matter.
Yours sincerely,
D. Reynolds, G. M. Monteiro, S. Ganeshan
List of changes
• Expanded the heuristic argument for the new coupling with a more in depth analysis of the microscopics.
• More clearly highlighted the aspects of our analysis which are unique to ferrofluids, giving a stronger connection to odd viscosity. This is done by splitting section 3 into two parts.
• Changed the title of the paper to “Parity breaking in….” instead of “3D Odd Viscosity in…”
• Changed brackets and Hamiltonian from velocity to momentum density, to garuntee the brackets satisfy the Jacobi identity. The incompressible limit is taken at the level of the equations of motion.
• Changed reference in abstract to a numbered citation.
• Restructured the third paragraph of the introduction to avoid mention of 3D active matter and more clearly state what we investigate.
• Added a new section reviewing the anisotropic viscosity tensor in 3D.
• Clearly indicated (with italics) the main results of each section, drawing attention to what is required for parity breaking in ferrofluids.
• Grammatical issues and typos in some inline equations.
• Formatting
Current status:
Reports on this Submission
Report
I thank the authors for their careful responses. However, there are still some points in the revised version that remain unclear:
1. The authors mention that the term "odd viscosity" is used broadly. Given the current logic of the manuscript, is the condition that the two paritybreaking odd coefficients in 3D are related because the authors aim to connect with 2D odd viscosity, or because they assume the system is Hamiltonian? Additionally, the authors invoke relativistic plasmas for some reason. In both relativistic and nonrelativistic plasmas, the relation in the collisionless limit holds as computed from kinetic theory. However, in both cases, collisions break this relation. I do not see a clear argument for why such a relation should hold in ferrofluids. This seems to relate to the previous criticism about whether describing hydrodynamics using Hamiltonians is a generic approach or not. The current discussion still suggests that the formalism imposes some constraints.
Another way to approach the problem is through kinetic theory. One could argue that Hall viscosities are nondissipative, allowing us to consider equilibrium distributions to understand them. However, this doesn't work because, despite being nondissipative, the coefficients are in front of gradients, meaning we need to move away from equilibrium to observe them. Somehow, the authors appear to move in the opposite direction by starting from a collisionless limit and drawing general conclusions about odd viscosities. I don’t see how this approach is correct. Specifically, if one uses different methods to arrive at hydrodynamic equations and the magnetizationvorticity coupling proposed in the paper, would the result be the same?
2. To clarify my question about magnetization: is the proposed transport coefficient for ferrofluids only nonzero in the quasihydrodynamic regime, where magnetization has not yet relaxed, or in the purely hydrodynamic regime? I don’t see how a field that has relaxed or been integrated out can still influence the transport coefficients. It would be beneficial to see more details included
Additionally, a minor question: is $M^0$ the same as $ \vec{M}^0 $?
Recommendation
Ask for major revision
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I do not understand reference [47] which tells the reader to contact Ferrotec for data on the hydro properties of ferrofluids. If this is straightforward then the author could have contacted Ferrotec, if it isn't then why give the reader the runaround? In all the articles and books cited in ferrohydrodynamics, did nobody ever bother to mention any numbers? If I check [34], I find that the first sentences discuss data on hydro properties of ferrofluids.
In the rebuttal reference [33] is mentioned which confirms the incompressible nature of ferrofluids. I do not understand why this is only stated in the rebuttal and not in the revised manuscript; this is such a central assumption in all of the computations and it is absolutely not obvious that this holds.
I sympathize with the idea that as a theorist one works in a more exploratory way where one is not necessarily bothered too much with the experimental realizability and that when asked about details on experimental realizability one replies with "we leave details of ferrofluid experiments to future work", however the nature of this work is such that it is composed of theoretical insights from previous works which are put together in a modified form with the claim that a very specific system can potentially confirm the corresponding theoretical results. In this case, I believe it is justified to ask this question as I believe this to be the bottom line.
Section IV is new and is seemingly added in part to address my comments about not discussing how the paritybreaking term in (34) can be understood as an entry in the viscosity tensor. However, I think some statements in relation to timereversal symmetry are incorrect or at least very misleading. It is implied in this section that shear viscosity breaks timereversal symmetry, just like odd viscosity does. This is not correct, it is only odd under T, which it should be since it is a diagonal dissipative coefficient. The fact that it is odd means it is covariant under timereversal. Shear viscosity occurs for fluids that are microscopically Tsymmetric, whereas odd viscosity requires intrinsic Tbreaking, meaning that Pbreaking fluids with PTsymmetry display both odd viscosity and shear viscosity. It cannot display terms that are truly PTbreaking, although in this specific case it is not possible to write down such a term. To properly constrain these terms requires accounting for the OnsagerCasimir relations.
I also think section IV is somewhat excessive, since the point of the preceding sections is to show how a specific odd entry in the viscosity tensor arises so why do we now need this general analysis based on symmetry and the second law. I also do not understand why after (38) shear viscosity is seemingly discarded ("So, instead of focusing on a general viscosity tensor, we will only focus on PT symmetric terms") but then it is brought back again in section VI. It seems that this section builds towards the conclusion "Therefore, in this case, referring to parityodd viscosity coefficients by 3D odd viscosity is only accurate upon the identification νo⊥ = −2νo∥." which is the same as saying that not all the odd viscosity coefficients allowed by symmetry are independent.
In summary, I think section IV is completely out of place in the manuscript. It saddens me that this is so, because as mentioned I specifically requested a discussion of the paritybreaking term as an entry in the viscosity tensor but this is definitely not what I had in mind.
Requested changes
 It says "in the in the"
 Remove section IV, but just explain how this paritybreaking entry in the momentum balance equation is a specific type of 3d odd viscosity
Provide more details on ferrofluids
Recommendation
Ask for major revision
Report
The resubmitted manuscript addresses all my previous questions and comments satisfactorily. The paper is wellwritten and presents new and interesting physics. I recommend publishing it in SciPost Physics in its current form.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)