Brownian noise eﬀects on magnetic focusing of prolate and oblate spheroids in channel ﬂow Mohammad Reza Shabanniya1and Ali Naji2a

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Brownian noise eﬀects on magnetic focusing of prolate and oblate

spheroids in channel ﬂow

Mohammad Reza Shabanniya1and Ali Naji2, a)

1)School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19538-33511,

Iran

2)School of Nano Science, Institute for Research in Fundamental Sciences (IPM), Tehran 19538-33511,

Iran

We investigate Brownian noise eﬀects on magnetic focusing of prolate and oblate spheroids carrying perma-

nent magnetic dipoles in channel (Poiseuille) ﬂow subject to a uniform magnetic ﬁeld. The focusing is eﬀected

by the low-Reynolds-number wall-induced hydrodynamic lift which can be tuned via tilt angle of the ﬁeld

relative to the ﬂow direction. This mechanism is incorporated in a steady-state Smoluchowski equation that

we solve numerically to analyze the noise eﬀects through the joint position-orientation probability distribu-

tion function of spheroids within the channel. The results feature partial and complete pinning of spheroidal

orientation as the ﬁeld strength is increased and reveal remarkable and even counterintuitive noise-induced

phenomena (speciﬁcally due to translational particle diﬀusivity) deep into the strong-ﬁeld regime. These in-

clude ﬁeld-induced defocusing, or lateral broadening of the focused spheroidal layer, upon strengthening the

ﬁeld. We map out focusing ‘phase’ diagrams based on the ﬁeld strength and tilt angle to illustrate diﬀerent

regimes of behavior including centered focusing and defocusing in transverse ﬁeld, and oﬀ-centered focusing

in tilted ﬁelds. The latter encompasses two subregimes of optimal and shouldered focusing where spheroidal

density proﬁles across the channel width display either an isolated oﬀ-centered peak or a skewed peak with a

pronounced shoulder stretching toward the channel center. We corroborate our results by analyzing stability

of deterministic ﬁxed points and a reduced one-dimensional probabilistic theory which we introduce to semi-

quantitatively explain noise-induced behavior of pinned spheroids under strong ﬁelds. We also elucidate the

implications of our results for eﬃcient shape-based sorting of magnetic spheroids.

I. INTRODUCTION

Magnetic separation of nano/microparticles in micro-

ﬂuidic setups have emerged as an important technological

tool especially in biomedical research1–6. The separation

methods typically involve a mixture of magnetic or mag-

netically labeled biological or synthetic particles in an

imposed shear ﬂow within properly designed microchan-

nels. The system is subjected to an external magnetic

ﬁeld to produce controlled alterations in the ﬂow-driven

motion of target particles based on their size, shape and

magnetic properties6–9. A host of bioparticles can be

manipulated using such setups. These include proteins,

nucleic acids, bacteria such as Escherichia coli, and cells

such as red blood cells and circulating tumor cells10–14.

Similar methods can be used for detection and sorting

of artiﬁcial magnetic beads. The latter often consist of a

nonmagnetic (e.g., spherical or ellipsoidal) core coated or

doped by magnetic materials (e.g., ferromagnetic chrome

dioxide, superparamagnetic magnetite, etc.)15–23.

Magnetic separation techniques often use spatially

nonuniform magnetic ﬁelds to create net magnetic forces

on the target particles. The force magnitude is tuned by

adjusting, e.g., the ﬁeld gradient and/or the magnetic

susceptibility mismatch between the particles and the

background ﬂuid, enabling cross-stream magnetophoretic

migration and, hence, separation of the particles24.

a)Corresponding author. Email: a.naji@ipm.ir

Utilizing uniform magnetic ﬁelds has also received

mounting attention in recent years as an alternative route

to separation of magnetic (nonmagnetic) particles in non-

magnetic (magnetic) microﬂuidic ﬂows25–35. In a uni-

form ﬁeld, the magnetophoretic eﬀect is absent, as mag-

netic particles experience no net force from the applied

ﬁeld but only a torque that augments the torque expe-

rienced from the imposed shear. These torques are both

inherently shape dependent. Thus, other than oﬀering

practical advantages (e.g., access to large ﬁeld-exposed

areas and high-throughput parallelization of multiple

channels26), uniform ﬁelds facilitate a robust shape-based

approach to magnetic particle separation5,6.

Recent experiments have indeed established the above

strategy for separation of prolate paramagnetic spheroids

from paramagnetic spheres of the same volume26,27.

These experiments utilize a pressure-driven channel ﬂow

at low Reynolds number and a uniform (static) mag-

netic ﬁeld applied either transversally or with a tilt an-

gle relative to the ﬂow direction. In the absence of

ﬁeld, the particles were shown to undergo standard Jef-

fery rotation36,37 due to the shear-induced torque. In

a transverse ﬁeld26, the up-down symmetry pertinent to

these rotations breaks down for prolate spheroids but not

the spheres (magnetic torque vanishes on paramagnetic

spheres). This rotational asymmetry is coupled with lat-

eral particle translation across the channel width through

the particle-wall hydrodynamic interactions, speciﬁcally,

the wall-induced hydrodynamic lift, which in turn ef-

fects net migration of prolate spheroids to the chan-

nel center. This mechanism holds under weak ﬁelds,

arXiv:2210.10918v1 [cond-mat.soft] 19 Oct 2022

also for nonmagnetic particles in ferroﬂuid ﬂows26,27. It

has been scrutinized by deterministic simulations and

theories26–28,31–35 which also illuminate the more com-

plex particle migration pattern observed27 under tilted

ﬁelds and the orientational pinning of spheroids26–34,38,39

that follows from the counterbalance between ﬁeld and

shear-induced torques at strong ﬁelds40.

The strong-ﬁeld mechanism for separation of mag-

netic spheroids with pinned orientation was ﬁrst ad-

dressed in Refs.29,30. These studies considered prolate

spheroids of permanent (ferro)magnetic dipole moment

in two-dimensional (2D) or plane Poiseuille ﬂow and also

in three-dimensional (3D) ﬂows within rectangular and

cylindrical channels. Using far-ﬁeld hydrodynamic cal-

culations of the wall-induced lift, they showed that ori-

entationally pinned spheroids laterally translate within

the channel and focus at speciﬁc shape-dependent lat-

itudes upon adjusting the ﬁeld direction. The far-ﬁeld

theory closely reproduces boundary element simulations

especially at spheroid-wall separations exceeding a couple

particle sizes. When the applied ﬁeld is strong enough,

particle translation and rotation largely occur within the

ﬁeld plane26,27,29,40, and the results in 3D rectangular

channels of suﬃciently large cross-sectional aspect ratio

reproduce those in the 2D ﬂow29,30. In this latter set-

ting, the diﬀerences between cross-stream migration of

prolate ferromagnetic and paramagnetic spheroids under

weak and strong magnetic ﬁelds have also been studied

using direct ﬁnite-element numerical simulations31.

In this paper, we also consider spheroidal particles of

permanent magnetic dipole in plane Poiseuille ﬂow under

a uniform magnetic ﬁeld (of arbitrary tilt angle within

the ﬂow plane) according to the models in Ref.29,31. In

the theoretical and numerical studies cited above, lat-

eral migration and focusing of (para/ferro)magnetic par-

ticles have mainly been analyzed based on deterministic

particle trajectories and, to our knowledge, noise eﬀects

have been examined only brieﬂy by Brownian Dynamics

(BD) simulations in Ref.29. Our goal here is to elucidate

salient, unexplored aspects of Brownian noise, or particle

diﬀusivity, which we show to have important qualitative

ramiﬁcations that go beyond the deterministic picture.

We use three complementary approaches and con-

sider not only prolate but also oblate spheroids that

have received much less attention41–44. The approaches

used here are (i) a continuum formulation based on

a steady-state Smoluchowski equation to describe the

joint position-orientation probability distribution (PDF)

of spheroids from weak to strong ﬁelds (without having

to deal with sampling ﬂuctuations of analogous BD sim-

ulations); this will be referred to as the full probabilistic

approach; (ii) a systematic phase-space stability analysis

to classify ﬁxed points of deterministic spheroidal dy-

namics at strong ﬁelds; (iii) a reduced probabilistic theory

to describe the noise-induced behavior of spheroids on

a one-dimensional (1D) pinning curve in the position-

orientation coordinate space at strong ﬁelds. To achieve

our goals, the far-ﬁeld hydrodynamic contributions (in-

cluding the hydrodynamic lift) due to the channel con-

ﬁnement need to be derived without imposing the strict-

pinning constraint of Ref.29 on the spheroidal orientation.

We provide these derivations in the Supplementary Ma-

terial (SM) and use them within approach (i) and also

within our linear stability analysis and determination of

the eigenvalues associated with ﬁxed points of the de-

terministic spheroidal dynamics in approach (ii). The

strict-pinning form of our generalized far-ﬁeld expressions

are used within approach (iii) and the nonlinear stability

analysis of higher-order ﬁxed points in approach (ii).

We show that the translational noise (particle dif-

fusivity) across the channel width can speciﬁcally im-

part unexpected eﬀects in the strong-ﬁeld regime which

is of main interest in this work. In particular, under

a transverse (longitudinal) ﬁeld, where prolate (oblate)

spheroids are focused at the channel center, we ﬁnd

that the translational noise causes a peculiar ﬂat-top

subGaussian density proﬁle for the spheroids across the

channel width. It also triggers a counterintuitive ﬁeld-

induced defocusing upon amplifying the applied ﬁeld be-

yond a certain threshold; that is, strengthening the ﬁeld

leads to continual broadening of the spheroidal layer well

within the strong-ﬁeld regime. The defocusing follows

because (1) the dynamical stability of centered focus-

ing turns out to be of nonlinear order (i.e., the ﬁrst

and second derivatives of the lift relative to lateral po-

sition within the channel vanish at its center); and (2)

the higher-order stability progressively weakens (i.e., the

nonvanishing third derivative also tends to zero) as the

ﬁeld is ampliﬁed. These features make the determinis-

tic focusing of spheroids at the channel center prone to

signiﬁcant noise-induced alterations. They also stand at

odds with the linear stability with a ﬁnite ﬁrst derivative

which is reported for the corresponding case in Refs.29,30.

When the applied ﬁeld is tilted relative to the ﬂow, the

spheroids are focused at an oﬀ-centered latitude within

the channel, reﬂected by a localized peak in their steady-

state density proﬁle across the channel width. While our

results in this case partly corroborate those in Ref.29,

they unravel the subtle nature of the hydrodynamic lift

at the channel center which we ﬁnd to be a half-stable

ﬁxed point under a tilted ﬁeld. The half-stability engen-

ders a broad noise-induced ‘shoulder’ in the spheroidal

density proﬁles. The shoulder stretches from the focusing

latitude to the channel center. It can be strong enough to

undermine the oﬀ-centered focusing peak. Since an opti-

mal focusing peak (e.g., for particle separation purposes)

is desired to be sharp and isolated, we map out compre-

hensive ‘phase’ diagrams based on the ﬁeld strength and

tilt angle to identify various regimes of magnetic focusing

and, in particular, to distinguish the regime of optimal

focusing from that with noise-induced density shoulders.

We show that the reduced probabilistic theory (ap-

proach iii) can provide a direct link between the deter-

ministic ﬁxed points and the full probabilistic results by

means of a virtual lift potential deﬁned along the pinning

curve. To our knowledge, the full probabilistic approach

y=H

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prolate

oblate

✓B

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FIG. 1. Schematic view of magnetic spheroids (not drawn to

scale) with permanent dipole moment malong their symme-

try axis in a channel of width Hsubject to plane Poiseuille

ﬂow uf(r)and uniform magnetic ﬁeld Bof tilt angle θB.

and also the deterministic analysis as we report here have

not been considered in the present context before. A

quadratic lift potential has however been shown to exist

in the vicinity of the oﬀ-centered focusing latitude via a

Langevin equation in Ref.29. Our derivation of the re-

duced theory shows that this insightful observation can

formally be extended to the whole pinning curve, leading

to useful (albeit semiquantitative) analytical predictions.

We discuss our model and methods in Sections II and

III, our results for prolate spheroids in transverse ﬁeld

in Section IV and for prolate/oblate spheroids in tilted

ﬁelds in Sections V-VII. The implications of our results

for shape-based sorting of spheroids are given in Section

VIII. Details of our generalized far-ﬁeld calculations and

deterministic analysis appear in the SM and details of

our reduced probabilistic theory appear in Appendix D.

II. MODEL

The magnetic particles in our model are taken as rigid,

passive and neutrally buoyant colloidal spheroids within

a long planar channel of width Hand rigid no-slip walls

at y= 0 and H; see Fig. 1. The channel is subjected to

a stationary plane Poiseuille ﬂow uf(r) = uf(y)ˆ

xwhere

r= (x, y, z)is the position vector within the channel, ˆ

is the unit vector in x-direction, and uf(y)= ˙γy (1−y/H)

is the velocity proﬁle in y-direction with maximum shear

rate ˙γ= 4Umax/H and ﬂuid velocity Umax >0. The im-

permeability of channel walls is modeled via a harmonic

steric potential which creates repulsive transverse forces

when the spheroids come into contact with them; see Ap-

pendix A. A static uniform magnetic ﬁeld Bis applied

within the x−yplane; i.e., B·ˆ

z= 0 where ˆ

z=ˆ

x×ˆ

yin

the out-of-plane unit vector. The ﬁeld strength and tilt

angle (relative to the ﬂow direction) are B=|B|and θB.

The spheroids are assumed to be of equal volume V(α)

and aspect ratio α=a/b, where ais half-length of the

spheroidal axis of symmetry, i.e., the major (minor) body

axis for prolate (oblate) shapes α>1(α <1), and bis half-

length of the other two axes. The spheroids carry per-

manent magnetic dipole moments m=mˆ

d(m≥0) along

their symmetry axis identiﬁed by the orientation unit vec-

tor ˆ

d. Diﬀerent shapes will be studied at constant volume

V(α) = V0, or eﬀective radius Reﬀ = (3V0/4π)1/3(radius

of reference sphere,α= 1) and, hence, constant dipole

moment26,27,29. Since the shear and ﬁeld tend to orient

the symmetry axis in the x−yplane, ˆ

dis assumed to vary

in-plane at strong enough shear/ﬁeld strengths26,27,29,40.

We can thus use the 2D parametrization ˆ

d=(cos θ, sin θ)

with θbeing the orientation angle relative to the x-axis.

III. CONTINUUM PROBABILISTIC FORMULATION

We assume that the translational and rotational dy-

namics of spheroids within the channel are governed by

deterministic forces and torques that result from the im-

posed shear, the external magnetic ﬁeld, the spheroid-

wall steric and hydrodynamic interactions, and also the

Brownian (thermal) ambient noise. Such a description

is expected to hold for suﬃciently dilute suspensions,

where interparticle forces are negligible, or for individual

spheroids advecting through the channel. The system

properties can then be studied using a continuum proba-

bilistic approach based on a noninteracting Smoluchowski

equation that governs the joint position-orientation prob-

ability distribution function (PDF) Ψ(r;ˆ

d)of spheroids

with center position rand axial orientation ˆ

d. In the

steady state, the Smoluchowski equation reads45,46

∇r·(vΨ) + Rˆ

d·(ωΨ) = ∇r·DT·∇rΨ + DRR2

dΨ,(1)

where Rˆ

d=ˆ

d× ∇ˆ

dis the rotation operator, ∇ˆ

dis

the unconstrained Cartesian gradient relative to ˆ

d, and

v=v(r;ˆ

d)and ω=ω(r;ˆ

d)are the net deterministic

translational and angular ﬂux velocities of spheroids.

In Eq. (1), DT(α)is the second-rank translational

diﬀusivity tensor and DR(α)is the in-plane rotational

diﬀusivity due to translational and rotational Brownian

noises, respectively. The translational diﬀusivity ten-

sor is written as DT(α) = D

(α)ˆ

dˆ

d+D

⊥(α)I−ˆ

dˆ

d

where Iis the identity tensor and D

(α)and D

⊥(α)are

the longitudinal and transverse diﬀusivities in parallel

and perpendicular directions relative to the spheroidal

symmetry axis. The diﬀusivities are assumed to fol-

low the Stokes-Einstein relations for no-slip spheroids.

They are expressed using spheroidal shape functions47–50

∆

,⊥(α)=D

,⊥(α)/D0T and ∆R(α)=DR(α)/D0R whose

explicit forms are reproduced in Section 1, SM. Here,

D0T =kBT/(6πηReﬀ )and D0R =kBT/(8πηR3

eﬀ )are

translational and rotational diﬀusivities of the reference

sphere, ηis the ﬂuid viscosity and kBTis the thermal en-

ergy. Also, DT(α)and DR(α)express bulk diﬀusivities of

free spheroids, as the conﬁnement eﬀects due to spheroid-

wall steric and hydrodynamic interactions are explicitly

included via deterministic ﬂuxes that we discuss next.

A. Translational ﬂux velocities

The net deterministic translational ﬂux velocity of the

spheroids can be written as the sum of three terms

v(r;ˆ

d) = uf(r) + u(st)(r;ˆ

d) + u(im)(r;ˆ

d),(2)

where ufis the advective ﬂux, and u(st) and u(im) are the

translational ﬂux velocities due to spheroid-wall steric

and hydrodynamic interactions, respectively. The steric

term u(st) follows from the wall potential; see Eqs. (A2)

and (A3), Appendix A. The hydrodynamic term u(im)

is derived using the singularity image method of Blake51

by approximating the far-ﬁeld low-Reynolds-number ﬂow

around a no-slip spheroid of center position rand ori-

entation ˆ

dwith that of a second-order stresslet tensor

S(r;ˆ

d)37. Relative to a given wall, and using Einstein

summation, the components of u(im) at position r0read

u(im)

i(r0;ˆ

d) = −1

8πη K(im)

ijk (r0,r)Sjk(r;ˆ

d),(3)

where Sjk and K(im)

ijk are the components of S(r;ˆ

and the third-order hydrodynamic Green function ten-

sor K(im)(r0,r)and i, j, k denote Cartesian coordinates

which we show interchangeably by {x, y, z}or {1,2,3}.

The term u(im)(r;ˆ

d)thus gives the ﬂux velocity from

hydrodynamic interactions of a spheroid with its wall-

induced singularity images. It involves the equal-point

Green function (self-mobility tensor) K(im)(r,r)which we

use on the ﬁrst-image level from Refs.52,53.

The explicit forms of Sand u(im) are derived in Section

2 of the SM for arbitrary prolate/oblate aspect ratio, ﬁeld

strength and tilt angle and, thus, extend the expressions

given in Ref.29 for strictly pinned prolate spheroids. Such

an extension is essential to our later linearized stability

analysis and also our probabilistic analysis where ﬁeld-

induced pinning is not imposed but obtained as a result.

B. Angular ﬂux velocities

The net deterministic angular ﬂux velocity of the

spheroids can also be written as the sum of three terms

ω(r;ˆ

d) = ωf(r;ˆ

d) + ωext(ˆ

d) + ω(im)(r;ˆ

d),(4)

where ωf=ωf(y, θ)ˆ

z,ωext =ωext(θ)ˆ

zand ω(im) =

ω(im)(y, θ)ˆ

zare the shear, ﬁeld and wall-induced hy-

drodynamic angular velocities, respectively. The shear

and ﬁeld terms in the signed magnitude of ω, i.e., ω=

ωf+ωext +ω(im), standardly follow as (Section 2.3, SM)

ωf(y, θ) = ˙γ

21−2y

H(β(α) cos 2θ−1) ,(5)

ωext(θ) = DR(α)

kBTmB sin (θB−θ).(6)

Here, β(α)=(α2−1)/(α2+1) is the Bretherton number54.

For prolate/oblate spheroids, one has 0<±β(α)<1.

The image term ω(im) is derived in Section 2.5 of the

SM and is (due to its lengthy expression) given in rescaled

form in Eq. (B1). While this term has not been consid-

ered before and is included in our probabilistic analysis,

it turns out to be small and of negligible impact on our

results relative to the other angular terms (Appendix B).

C. Dimensionless representation

Due to translational symmetry in x-direction, the solu-

tions of Eq. (1) admit the form Ψ=Ψ(y, θ). The latitude-

orientation plane y−θis then taken as the reduced coordi-

nate space. Also, due to ﬂuid incompressibility, ufdrops

out of the l.h.s. of Eq. (1) and only the y-components of

u(st) and u(im) prevail, combining into the net transverse

velocity vy=u(st)

y+u(im)

ywhere u(im)

yis the hydrodynamic

lift. We rescale the length and time units with Reﬀ and

the inverse rotational diﬀusivity D−1

0R , respectively, lead-

ing to the rescaling of lateral coordinate and ﬂux veloc-

ities as ˜y=y/Reﬀ ,˜vy(˜y, θ) = vy(Reﬀ ˜y, θ)/(Reﬀ D0R)and

˜ω(˜y, θ) = ω(Reﬀ ˜y, θ)/D0R. The dimensionless parameter

space is thus spanned by the aspect ratio α, tilt angle

of the ﬁeld θB, rescaled channel width ˜

H, ﬂow Péclet

number (rescaled shear strength) Pef, and the magnetic

coupling parameter (rescaled ﬁeld strength) χ, where

H=H

Reﬀ

,Pef=˙γ

D0R

and χ=mB

kBT.(7)

We obtain the rescaled forms of the net transverse ve-

locity, the hydrodynamic lift and the net angular velocity

(see Appendices Aand Band Section 2 of the SM) as

˜vy(˜y, θ) = ˜u(st)

y(˜y, θ) + ˜u(im)

y(˜y, θ),(8)

˜u(im)

y(˜y, θ) = −ζ(α)1

˜y2−1

(˜

H−˜y)2sin 2θ(15Pef

64 1−2˜y

H−XM+ 2YM−ZMcos2θ+2XM−2YMsin2θ

−9β(α)

16 1

2YHPef1−2˜y

Hcos 2θ+χ

ζ(α)sin (θB−θ)),(9)

˜ω(˜y, θ) = Pef

21−2˜y

H(β(α) cos 2θ−1) + χ∆R(α) sin (θB−θ) + ˜ω(im)(˜y, θ),(10)

where XM, Y M, ZMand YHare shape-dependent resis-

tance functions (see Table 1 of the SM), and ζ(α) = α2

and α−1for prolate and oblate spheroids, respectively37.

Our expression in Eq. (9) recovers the standard zero-

lift result, u(im)

y= 0, for nonmagnetic spheres (χ= 0;

α=1,XM=YM=ZM=1,YH= 0) due to the absence of

inertia in the present setting54–60. It also shows that the

lift is consistently zero on (ferro)magnetic spheres (χ >0)

under a uniform ﬁeld of arbitrary tilt angle. For this rea-

son, spheres are excluded from our later discussion given

their lack of magnetic focusing in the present context.

D. Rescaled Smoluchowski equation

The rescaled form of the Smoluchowski equation (1)

for the joint latitude-orientation PDF of spheroids reads

∂

∂˜y˜vy(˜y, θ)˜

Ψ+∂

∂θ ˜ω(˜y, θ)˜

Ψ=(11)

3∆+(α)−∆−(α) cos 2θ∂2˜

∂˜y2+ ∆R(α)∂2˜

∂θ2,

where ∆±= (∆

±∆

⊥)/2combine the shape functions

∆

,⊥. With no loss of generality, we choose the com-

putational domain to solve Eq. (11) as y∈[0, H]and

θ∈[θB−π, θB+π)(we may later depict the PDF over

other visually suitable intervals of θ). The rescaled PDF

is deﬁned as ˜

Ψ(˜y, θ) = Reﬀ Ψ(Reﬀ ˜y, θ)/¯nwhere ¯n=

0RθB+π

θB−πΨ(y, θ) dθdygives the mean number of parti-

cles per unit x−zarea in a 3D realization of the system;

hence, the normalization R˜

0RθB+π

θB−π˜

Ψ(˜y, θ) dθd˜y=1.

Equation (11) is solved numerically using ﬁnite-

element methods and by imposing periodic boundary

conditions on θand, for numerical convenience, no-ﬂux

boundary conditions on the channel walls50,61. We vary

the system parameters in ranges that cover realistic val-

ues (Appendix C). The tilt angle of applied ﬁeld is re-

stricted to the ﬁrst quadrant 0≤θB≤π/2to avoid re-

dundant solutions, as solutions of Eq. (11) in other θB-

quadrants can be recovered using the symmetry trans-

formation {θB→θB+π, θ →θ+π}39,40 and, as our

numerical inspections for the key regimes of interest here

indicate, also using {θB→π−θB, θ →π−θ, ˜y→˜

H−˜y}.

E. Analytical schemes complementing numerical solutions

In our numerical implementation of Eq. (11), i.e., the

full probabilistic approach, we retain all ﬂux velocities in

Eqs. (8) and (10). In the key regimes of magnetic fo-

cusing analyzed in this work (Sections IV and V), the

spheroids are focused suﬃciently away from the walls.

This renders the steric term ˜u(st)

yirrelevant. As noted

before, the wall-induced hydrodynamic angular velocity

˜ω(im) also stays negligible (Appendix B). Discarding the

above two terms enables approximate analytical schemes

that can be used to further illuminate our numerical ﬁnd-

ings. Since we frequently allude to the predictions of

these schemes in the upcoming sections, a summary of

their scope and applicability will be in order:

•Deterministic dynamical stability analysis: This ap-

proach is based on noise-free translational and rota-

tional dynamical equations for the spheroidal motion,

˙y= ˜vy(˜y, θ)'˜u(im)

y(˜y, θ),(12)

θ= ˜ω(˜y, θ)'˜ωf(˜y, θ) + ˜ωext(θ),(13)

where ˜

θ=ˆ

θ·(˜

ω×ˆ

d)and ˆ

θis the counterclockwise polar

unit vector. This approach is useful in predicting the

boundaries of most, albeit not all, regimes of magnetic

focusing that we later identify in this work. The said

boundaries are determined based on the dynamical

stability of ﬁxed points (˜y∗, θ∗)of Eqs. (12) and (13)

which follow by setting ˜

˙y=˜

θ=0. The ﬁxed points can

be obtained from the intersections of nullclines that

give the loci of coordinate-space points where either

˙yor ˜

θvanishes62–64. According to Eq. (12), ˜

˙y= 0

yields the zero-lift nullclines. From Eq. (9), these fol-

low as the channel centerline ˜y=˜

H/2and the lines

θ=nπ/2for integer n. On the other hand, according

to Eq. (13), ˜

θ= 0 yields the loci of coordinate-space

points where the shear and ﬁeld-induced torques coun-

terbalance and produce zero net angular velocity. The

ensuing nullclines include the orientationally stable

(deterministic) pinning curve θ=θp(˜y)along which

∂˜

θ/∂θ < 0and, hence, the spheroidal orientation is

pinned29,30,40. We shall elaborate on this later.

While deterministic equations have previously been

used to study noise-free trajectories of prolate

spheroids29,30, a comprehensive study to classify the

phase-space dynamical stability of their ﬁxed points

has been missing. We present such an analysis for

both prolate and oblate spheroids in Section 3 of the

SM and use its predictions in what follows to interpret

the probabilistic solutions obtained from Eq. (11).

While the deterministic results are valuable, they ex-

hibit important departures from the probabilistic ones

especially at strong ﬁelds, contrasting the intuitive as-

sumption that noise (particle diﬀusivity) can be ne-

glected in the strong-ﬁeld regime29,30.

For future reference, the ﬁxed point stability is stan-

dardly determined on the linearization level by the

eigenvalues λyand λθof the Jacobian matrix

J=∂(˜

˙y, ˜

θ)

∂(˜y, θ)'



∂˜u(im)

∂˜y

∂˜u(im)

∂θ

∂˜ω

∂˜y

∂˜ω

∂θ 

˜y∗,θ∗

.(14)

When λyand/or λθvanish, we use a higher-order

(nonlinear) analysis62–64 to determine the true nature

of the ﬁxed point. For the most part, λyand λθturn

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BrowniannoiseeectsonmagneticfocusingofprolateandoblatespheroidsinchannelowMohammadRezaShabanniya1andAliNaji2,a)1)SchoolofPhysics,InstituteforResearchinFundamentalSciences(IPM),Tehran19538-33511,Iran2)SchoolofNanoScience,InstituteforResearchinFundamentalSciences(IPM),Tehran19538-33511,IranWeinvesti...

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Brownian noise eﬀects on magnetic focusing of prolate and oblate spheroids in channel ﬂow Mohammad Reza Shabanniya1and Ali Naji2a

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