Shear-induced assembly and breakup in suspensions of magnetic Janus particles with laterally shifted dipoles

Abstract

We employ Brownian dynamics simulations to investigate the shear-induced assembly and breakup of aggregates in dilute suspensions of magnetic Janus particles with laterally shifted dipoles. By systematically displacing the magnetic dipole from its geometric center, given by the dipolar shift s, and the strength of magnetic interactions relative to flow- and Brownian-induced forces, given by the Péclet number Pe and the dipolar coupling constant λ, respectively, distinct aggregation regimes are revealed. At low dipolar shifts (s ≤ 0.1) and low Pe, shear-enhanced diffusion promotes particle collisions, leading to faster aggregation of particles forming loop-like clusters that align with the flow. As Pe increases, these structures fragment into smaller aggregates and eventually disperse into gas-like arrangements. In contrast, particles with medium dipolar shifts (s ≥ 0.2) exhibit significant stability, forming compact vesicle- and micelle-like assemblies that resist shear-induced breakup even at high Pe, provided λ is sufficiently large. Orientational analysis indicates that particles maintain head-to-tail, head-to-side, and antiparallel alignments under shear, depending on s and Pe. The critical Pe required to induce cluster breakdown increases with both s and λ, underscoring the stabilizing influence of lateral dipole displacement and strong magnetic interactions. The transition to gas-like dispersion occurs when hydrodynamic and Brownian torques on the particles overcome the torques resulting from the interparticle interactions. Overall, these findings provide fundamental insights into the non-equilibrium self-assembly of anisotropic colloids, offering a framework for designing advanced materials with tunable structural and dynamic properties in microfluidics, drug delivery, and magnetorheological applications.

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1 Introduction

Janus particles, possessing two distinct hemispherical functionalities, have emerged as versatile building blocks in colloidal science due to their ability to self-assemble into ordered structures with tunable properties.¹ Incorporating magnetic domains into these particles further enhances their potential; such magnetic Janus particles can be guided and manipulated via external fields, enabling a wide array of targeted colloidal processes.^2,3 Their anisotropy is particularly valuable in applications spanning drug delivery, where surface functionalities facilitate selective binding and controlled release;^4,5 microfluidics, which leverage magnetic responses for precise sorting and flow control;^6,7 and magnetorheological fluids and sensors, where tunable aggregation under field or flow conditions creates pathways for advanced material design.^8–19

A key facet of magnetic Janus particles is their ability to displace magnetic dipoles, thereby introducing greater diversity in the potential energy landscapes.^20–22 Centered dipoles commonly yield chain-like or ring-like assemblies, whereas the direction of dipole displacement (lateral or radial) in magnetic Janus particles can generate more intricate structures. For laterally displaced dipoles, Monte Carlo and Brownian dynamics simulations reveal a transition from chain formations to compact vesicles and micelles.^23–25 For radially shifted dipoles, simulations and experiments demonstrate the preferential formation of chains and close-packed clusters, as well as triangular lattice configurations, despite edge-induced deviations.^26–28 Both configurations enhance directional binding and establish a competition between interparticle magnetic attraction and field-induced alignment; however, lateral shifts favor curved geometries, whereas radial shifts promote compact arrangements.

Although extensive work has characterized the behavior of magnetic Janus particles in quiescent conditions^29,30 and in external magnetic fields,^31,32 relatively little is known about how these anisotropic particles respond to shear. Earlier research has demonstrated that shear flow can induce ordering in diverse isotropic colloidal systems,^33–37 where dynamics arise from the interplay of Brownian motion, hydrodynamic forces, and interparticle interactions.^35,38 In dilute suspensions, these effects govern aggregation,³⁹ while in denser systems, hydrodynamic interactions become more significant, influencing both the microstructure and rheological properties.⁴⁰ At low shear rates, Brownian forces tend to dominate, promoting local diffusion and driven clustering; however, as the shear rate increases, shear-induced forces can fragment aggregates and even disperse them completely.⁴¹

This relative competition between these two mechanisms can be represented by the Péclet number Pe. The complexity of these interactions under shear conditions plays a pivotal role in orienting anisotropic particles, such as amphiphilic Janus particles.^42–44 Studies incorporating magnetic behavior into anisotropic colloids that combine shear flow with magnetic fields suggest that viscous torques can disrupt or enhance particle alignment, highlighting that even simple flow fields can give rise to complex rearrangements in paramagnetic Janus suspensions.^45–50 However, studies dedicated to magnetic Janus particles featuring lateral dipolar shifts predominantly focus on quiescent conditions,^23,25,27,51 despite the potential significance of shear-driven microstructural changes in applications such as magnetorheological fluids and flow-based sensing.^52,53

Magnetic dipole–dipole interactions are known to increase relative to Brownian relaxation of the particles, characterized by the dipolar coupling constant λ. It is anticipated that shear flow will become less significant in disrupting these colloidal structures. The fundamental question of how off-centered dipoles rearrange and fracture under increasing shear has not yet been comprehensively addressed.⁴⁵

In this work, Brownian dynamics (BD) simulations are utilized to elucidate how laterally shifted magnetic dipoles influence shear-induced self-assembly and breakup in dilute suspensions of magnetic Janus particles. By systematically varying the position of the magnetic dipole away from the geometric center of the particle, λ, and Pe, we explore the formation and breakup of colloidal structures, examining how these parameters modulate cluster size, morphology, and orientational order of particles in each aggregate. The findings of this study pave the way for the design of responsive and flow-stable colloidal materials in which magnetic anisotropy can be harnessed to achieve robust or switchable aggregation.

The remainder of this paper is structured as follows. Section 2 details the modeling framework and key simulation parameters. Section 3 presents and discusses results related to cluster formation, orientational distributions, and the critical conditions under which hydrodynamic flow predominates over magnetic interactions. Finally, Section 4 summarizes the principal insights and outlines future directions for the control of anisotropic colloids in nonequilibrium environments.

2 Problem formulation

2.1 Model system

The model system consists of a suspension of N magnetic Janus spherical colloidal particles with a radius a, dispersed within a Newtonian incompressible fluid characterized by viscosity η, density ρ, and absolute temperature T, and subjected to an imposed shear flow. As depicted in Fig. 1, each particle accommodates a permanent dipole. These dipoles, identified as m_i and m_j for a pair of particles i and j, are laterally displaced by a distance S from the particle center. This displacement establishes the dipolar shift, expressed as s = S/a, where s = 0 signifies a dipole located at the particle's center and s = 1 denotes a dipole situated on the surface.

Fig. 1 Model system. A schematic representation of magnetic Janus particles under shear in a dilute colloidal dispersion. Particles with radius a are modeled as spheres with a laterally displaced permanent dipole. These dipoles, m_i and m_j, are offset by a distance S from the center of the particle. The translational velocity U_i describing the particle motion through the flow field, while the fluid velocity U^∞ = [small gamma, Greek, dot above]yê_x represents the shear flow, with [small gamma, Greek, dot above] as the shear rate. The local reference frame (x′, y′, z′), and the inertial frame (x, y, z) are indicated, along with the angular velocity Ω_i characterizing particle rotation.

The particles exhibit linear motion, with the translational velocity U_i describing their displacement through the fluid.⁵⁴ The rotation of each particle is characterized by its angular velocity Ω_i. The local reference system of each particle is represented as (x′, y′, z′), while the inertial reference frame of the simulation is (x, y, z). The imposed fluid velocity is defined by U^∞ = [small gamma, Greek, dot above]yê_x, where [small gamma, Greek, dot above] quantifies the rate at which the velocity in the flow direction along the x axis varies linearly with the gradient direction along the y axis. The fluid layers undergo deformation, which affects the translational and rotational dynamics of the dispersed particles due to the shear stress generated within the suspension. Consequently, the particles experience both rotational and translational motion as a result of hydrodynamic, magnetic, and Brownian contributions.

The model is restricted to a dilute limit with a volume fraction ϕ = 0.001, which permits the neglect of hydrodynamic interactions, particularly in dilute systems where their influence is minimal. This validation supports the application of the BD method to reduce computational costs by concentrating solely on magnetic dipole interactions, shear flow, and stochastic influences. In this context, we consider only the viscous drag resulting from fluid friction and the driving force exerted by the applied shear flow.

2.2 Governing equations

The motion of the magnetic Janus particles is modeled by numerically solving the translational and rotational overdamped Langevin equations:

F^H + F^P + F^B = 0 (1)

and

T^H + T^P + T^B = 0, (2)

where F^H and T^H are the viscous drag forces and torques exerted on the magnetic Janus particles due to fluid motion, F^P and T^P denote the net interparticle forces and torques resulting from pairwise magnetic and steric interactions, and F^B and T^B represent the stochastic effects of Brownian motion.³⁹ According to Stokes' law, the viscous drag force and torque acting on an isolated particle are given by:

F^H = R_U(U^∞ − U_i) (3)

and

T^H = R_Ω(ω^∞ − Ω_i), (4)

where R_U = 6πηa and R_Ω = 8πηa³ are the translational and rotational drag coefficients, respectively. The fluid velocity U^∞ is evaluated at the center of mass of the particle i, and represents the local vorticity of the fluid.^46,55

The interparticle interactions, which involve forces and torques arising from the potential energy between colloidal particles, are characterized by magnetic attraction and steric repulsion. The second terms of eqn (1) and (2) are defined by:

F^P_ij = −∇(m_i·B_ij) + F^R_ij, (5)

T^P_ij = −m_i × B_i + s_i × [−∇(m_i·B_ij)], (6)

where B_ij is the magnetic field generated by the dipole i at the location of the dipole j. The magnetic dipole–dipole interaction Φ between two particles is described by

r_dij = r_i − r_j; r_dij = |r_dij|,

where μ_o is the magnetic permeability of free space, and r_dij is the distance between the interacting dipoles, as illustrated in Fig. 1.^56,57 Furthermore, F^R_ij is the steric repulsion force that acts exclusively within a layer of thickness δ surrounding the spherical particle. This generates the repulsive effect that stabilizes the minimum distance between colloidal particles.⁵⁸ This force is derived from the Weeks–Chandler–Andersen potential:⁵⁹where ε represents the strength of the surface potential, r_ij = |r_ij| denotes the distance between the centers of mass, as illustrated in Fig. 1.

The components F^B and T^B in eqn (1) and (2) introduce stochastic elements into the system that are modeled using a Gaussian distribution. The fluctuation–dissipation theorem governs the Brownian motion of particles within the surrounding fluid. This theorem ensures that the random forces and torques exerted on the particles are consistent with the intrinsic properties of the fluid.

The particle evolution is obtained by integrating eqn (1) and (2) using the Euler–Maruyama method with a specified time step. This evolution can be discretized in a dimensionless form as follows:

where r* and Ω* represent the position and orientation of the particles during the time step Δt, satisfying τ_p ≪ Δt ≪ τ_D, where τ_p = 1/R_U is the inertial relaxation time and τ_D = a²/D₀ is the characteristic time for Brownian diffusion, with D₀ representing the translational diffusivity of an isolated colloidal particle. This ensures adequate resolution of Brownian dynamics while neglecting inertial effects.³⁹ The contributions of Brownian motion are characterized by statistical properties: zero average 〈Δr*^B〉 = 0, 〈ΔΩ*^B〉 = 0 and variance 〈Δr*^BΔr*^B〉 = 2IΔt, 〈ΔΩ*^BΩ*^B〉 = 3/2IΔt for Brownian steps and rotations, respectively.

To obtain improved tracking in three-dimensional space, the rotation of the particle dipoles was implemented using the quaternion scheme.⁶⁰ All variables in the eqn (9) and (10) are dimensionless based on the following criteria: particle position is scaled in terms of particle size (r ∼ a), time is scaled using the characteristic time for Brownian diffusion (t ∼ τ_D), and forces and torques are scaled by the characteristic Brownian force (F ∼ k_BT/a) and torque (T ∼ k_BT), respectively. The dipolar coupling constant λ = μ₀|m|²/(4πk_BT) represents the strength of the magnetic dipole–dipole interactions relative to Brownian motion, thereby influencing cluster stability and the response to thermal fluctuations.^46,57 The Péclet number (Pe = [small gamma, Greek, dot above]a²/D₀) denotes the ratio between characteristic convective and diffusive timescales, which are critical for understanding the balance between shear forces and diffusion. This dimensionless number quantifies the extent to which shear flow dominates over Brownian motion.⁴⁰

2.3 Simulated parameters

The initial positions and orientations of the magnetic Janus particles were randomly assigned within a periodic simulation box. The implementation of Lees–Edwards boundary conditions was employed to account for shear flow effects, thereby enabling periodic shear flow throughout the simulation box. Within this framework, particles traversing one boundary of the simulation box re-enter through the opposing boundary, with their velocities being adjusted to precisely mirror the imposed velocity gradient. Such conditions are particularly well-suited for the simulation of unbounded shear flow, effectively mitigating the influence of boundaries on the interactions between particles.

The dipolar shift s was varied from 0.0 to 0.5 according to previous studies, where values that approximated s ∼ 0.56 were found to reproduce Fe₃O₄ shifted-dipole configurations observed experimentally.²² For 0.5 < s ≤ 0.8, parameters were extrapolated following the established trend fit (R² ≥ 0.95). The numerical stability of the simulations s > 0.5 became increasingly sensitive to shear flow effects as the dipole approached the surface of the particle. This necessitated the use of shorter time steps, consistent with the challenges noted in systems undergoing more significant transitions.²³ A time step Δt of up to 10⁻⁶ was employed to address these numerical instabilities. Pe was varied across the range 0 ≤ Pe ≤ 10³. For shifts s < 0.5 and Pe < 100, the time step was maintained at 10⁻⁴ but was decreased for simulations where Pe exceeded 100. This configuration was selected to optimize computational resources while ensuring that the results accurately captured the dynamics, as it has been demonstrated to influence nucleation and growth in dilute systems.^23,27

The number of particles, N, was selected to obtain a statistically representative sample, with system size effects examined for particle counts ranging from 250 to 1000 for s < 0.2. Meanwhile, systems with 0.2 < s ≤ 0.5 utilized between 250 and 500 particles, as system size effects were found to be negligible. This choice is consistent with previous simulation studies on dipolar colloids,^23,27 where systems with N ≳ 200 were found to provide statistically reliable data under directional aggregation conditions. In our case, the presence of shear flow imposes a directional bias that reduces configurational randomness and enhances dipolar alignment, effectively increasing the strength of interparticle interactions. Based on this reasoning and supported by analysis of system size independence in cluster size distributions, this number of particles was selected as a representative of a computationally efficient system size. This analysis was presented in the S1 (ESI†). Each system was simulated five times; the duration of the simulation was carefully determined to ensure statistical robustness. To establish statistically steady-state temporal conditions, we monitored the temporal evolution of cluster properties (as defined in Section 2.4). We considered statistical independence to be achieved when all measured quantities exhibited steady fluctuations with less than <5% variation over 200 consecutive time units, a criterion that was applied consistently across all five replicates. The dipolar coupling parameter λ = 15–60 encompasses ranges that capture the transition from the onset of dipole clustering (λ > 10) to strong-coupling magnetic Janus-relevant regimes, thereby avoiding simulation errors caused by finite-size effects (λ > 75). This range captures both the self-assembly of off-center dipoles and field-responsive collective states relevant to magnetic Janus particles.^23,24,27

2.4 Cluster properties

The identification of particle clusters in our simulations was based on an interaction criterion that employed a dimensionless bounding factor. Particles are considered irreversibly bonded if their center-to-center distance lies within a binding radius (r_b) defined by the interaction potential. For dipolar magnetic particles, this distance scales as r_b ∼ λ^1/3. For this model system, the threshold is r_ij ≤ 1.2·(2a),^23,27,61 which satisfies r_b < λ^1/3 for all conditions considered in this work. This threshold minimized erroneous quantification and facilitated precise differentiation between cluster arrangements. This approach was validated through statistical analyses, providing a robust and accurate characterization of cluster formation. The influence of Lees–Edwards boundary conditions on cluster identification was addressed using the unfolding method.^62,63

Subsequent to the identification of clusters, we conducted a quantification of all clusters c present within the system. We counted both the total number of clusters N_c and the number of constituent particles in each cluster p, N_c,p. Utilizing this data, we calculated the weight-averaged mean cluster size 〈N_c〉 by applying the following equation:

To quantify the orientational ordering of the particles, the bonded particle orientation distribution function P(m_i·m_j) was determined. This calculation considered two particles to be bonded if they satisfied the previously defined distance criterion, thereby facilitating comprehensive insights into the preferred orientations and arrangements of particles within clusters under varying shear conditions. The effective cluster radius 〈R_eff〉 was also calculated, which is defined as the average distance from the cluster center of mass to the most distant particle in the cluster, thereby capturing the spatial extent of the aggregate.

3 Results

3.1 Shear-induced cluster dynamics

The structural evolution of systems under shear flow is influenced by variations in Pe and s, which play a critical role in the formation and stability of particle assemblies. Fig. 2 presents the steady-state configurations (t/τ_D ∼ 1500) in the flow-velocity gradient plane xy for the values of the dipolar shift s and Pe, in a regime characterized by a high dipolar coupling constant (λ = 45) and a volume fraction of ϕ = 0.001. The images highlight microstructures that are driven by shear. The ground state structures identified in previous studies and categorized by s conform to the classification system established by Vega-Bellido et al.²³

Fig. 2 Simulation renderings of a dilute system of magnetic Janus particles at λ = 45 and dipolar shifts, low values (s = 0.0–0.1) and medium values (s = 0.2–0.5), at steady-state simulations times, projected on the flow and gradient plane xy under shear. Panels (a), (e), (i), (m) and (q) show snapshots at Pe = 0. Panels (b), (f), (j), (n) and (r) show overviews with significant changes in clusters formed at Pe = 1. Panels (c), (g), (k), (o) and (s) show a rearrangement in structures at Pe = 30. Gas-like structures are observed in panels (d), (h), (l) and (p) at Pe = 300. Finally, panel (t) shows compact clusters at s = 0.5.

At low s, this type of system primarily forms long chains and ring-like structures under quiescent conditions, as shown in Fig. 2(a) and (e). At Pe = 1, Brownian motion predominates; however, the shear force promotes the rearrangement of the particles into loop structures aligning with the flow direction (Fig. 2(b) and (e)). As Pe increases, the torque-induced flow overcomes magnetic dipolar interactions,⁵⁷ disrupting alignment and fragmenting structures into smaller chains and quadruplets (rectangular elongated arrangements) as shown in Fig. 2(c) and (g). At Pe = 300, the system transitions into a gas-like state, wherein the particles remain predominantly dispersed, as illustrated in Fig. 2(d), (h), (l) and (p)).

At medium s, the increased asymmetry in magnetic interactions and Brownian forces leads to the formation of compact clusters, including vesicle-like structures (Fig. 2i), amorphous ring micelles (Fig. 2m), and triplets (see Fig. 2q). At Pe ≤ 1, shear forces are insufficient to disrupt magnetic aggregation but are capable of organizing particles into complex configurations (Fig. 2(j), (n) and (r)). At Pe > 1, shear forces induce torques that reorganize particles into quadruplets, amorphous rings, and micelles elongated in the direction of flow (Fig. 2(k), (o) and (s)). It should be noted that an interesting feature is observed in s = 0.4–0.5: clusters exhibit exceptional stability under shear stress, maintaining their structure even at high Pe, as shown in Fig. 2(n), (o), (r) and (s), suggesting that moderate dipolar shifts enhance resistance to deformation. In fact, even at high shear intensity, configurations such as micelles and triplets remain stable under the specific conditions of s = 0.5 and Pe = 300 (Fig. 2t), underscoring the critical role of magnetic interactions in counteracting shear-induced fragmentation.

Expanding on the preceding observations, a quantitative analysis of the weighted average cluster size 〈N_c〉, as defined in eqn (11), over a temporal scale reveals distinct dynamical regimes governed by the interplay between dipolar interactions and shear forces (see Fig. 3(a)–(d)). At low s and (Pe ≤ 1) (open symbols), 〈N_c〉 exhibits time-dependent growth resembling diffusion-limited aggregation (DLA, where cluster growth is controlled by particle diffusion) as observed in quiescent systems,²³ evidenced by the characteristic power-law scaling (〈N_c〉 ∼ t^z) (Fig. 3(a) and (b)). While preserving this DLA-like behavior, weak flow induces two key modifications: (i) an earlier onset of aggregation compared to Pe = 0, and (ii) accelerated cluster growth due to shear forces that enhance particle collisions (Fig. 3(a) and (b)).

Fig. 3 Time-dependent aggregation of a dilute system of magnetic Janus particles at λ = 45. Panels (a)–(d) displays the weighted-averaged mean cluster size 〈N_c〉, as a function of time for values of s and Pe. Schematic representation in panel (a) show fragmentation scheme of cluster under Pe > 1. Panel (c) and (d) show zoomed-in view of the long-time dynamics (t/τ_D > 1000). The solid lines represent the power-law aggregation behavior t^z. Filled symbols correspond to time-independent systems, while open symbols indicate time-dependent systems. Of particular note is at Pe = 300 (open triangles in panels (a)–(c)), contrasting with the aggregation behavior at lower Pe. This exception is further analyzed in Fig. 4.

As Pe increases, 〈N_c〉 systematically decreases for all s, an effect further amplified in the medium s. This limitation on cluster growth is attributable to particles that are pushed together along the flow direction while being further separated along the velocity gradient direction, as shown in the scheme in Fig. 3a, which restricts aggregation primarily to the flow axis.³⁸ For intermediate values of s, the additional torque induced by magnetic dipolar displacement counteracts the shear-induced rotation. Consequently, clusters achieve metastable size saturation over extended timescales.⁶⁴ These stabilized structures maintain their dimensions by balancing dipolar alignment with shear stresses, thereby explaining both the reduced 〈N_c〉 and the emergence of flow-dependent steady states (see Fig. 3(c) and (d)).

3.2 Orientational distribution for observed cluster

The orientational analysis, which characterizes the alignment of dipolar particles, was performed using data from the final 20% of the simulation, representing the statistically steady-state simulation times. This approach thoroughly characterizes the predominant dipole orientations and cluster ordering. To quantitatively differentiate the cluster structures observed in Fig. 2, the orientation distribution function P(m_i·m_j) and the effective radius 〈R_eff〉 were calculated, as described in Section 2.4, for values of s ranging from 0.0 to 0.5, Pe (0, 1, 20, 30, 300) and λ = 45, as illustrated in Fig. 4. To complement the orientation analysis, the effective radius as a function of 〈N_c〉 during the aggregation process was employed as a metric to measure the spatial extent of the clusters under varying shear conditions, as shown in Fig. 4(a)–(e).

Fig. 4 Effect of steady shear on orientational distribution function and extended spatial radius for the structures and dipolar shift observed in Fig. 2. All at a fixed magnetic coupling constant of λ = 45. Panels (a)–(e) correspond to the R_eff as a function of 〈N_c〉. Panels (f)–(k) show the bonded particle orientation distribution function, P(m_i·m_j). Panels (l)–(q) contain the characteristic clusters for each of the systems.

At low s and Pe ≤ 1, the system exhibits a predominant peak at P(m_i·m_j) at m_i·m_j = 1, corresponding to the angle 0° (Fig. 4(f) and (g)). This alignment is energetically favorable for loop-like structures, as parallel dipoles minimize magnetic repulsion while facilitating curvature closure (see Fig. 4(l) and (m)), consistent with the head-to-tail configuration identified in prior research.⁵⁰ As Pe increases (Pe = 20, 30), loops fragment into smaller aggregates, and the range of peaks at m_i·m_j between 0–0.5 (see Fig. 4(g)) indicates a transition to head-to-side orientations. Geometrically, this reflects elongated rectangular arrangements in which the dipoles align at angles 60°–90° to compensate for shear-induced distortion and magnetic dipolar attraction. At higher Pe, gas-like structures arise, as illustrated in Fig. 4(h), (l), (m) and (n); although residual parallel alignment persists, emphasizing the dominance of magnetic interactions over shear.

In medium s and Pe ≤ 1, m_i·m_j exhibits peaks between 0.5 and 1 (see Fig. 4(h)), corresponding to the dipole angles of 0° to 60°. These orientations are characteristic of vesicle-like structures at s = 0.2 (Fig. 4(n)). As the shear effect increases, the primary peak shifts toward smaller values, m_i·m_j at 0 and 0.5, corresponding to the formation of small loop and quadruplet structures, as observed in Fig. 4(h) and (n). For s ≤ 0.3, the orientational peaks P(m_i·m_j) at 0, −0.5, and −1 indicate head-to-side alignments (90°), triangular (120°) and antiparallel dipole (180°), respectively (see Fig. 4(i) and (k)). These configurations are characteristic of amorphous rings, micellar, and triplet structures, as shown in Fig. 4(o) and (p), where the dipoles optimize packing efficiency while minimizing steric repulsion.²³ Even under Pe ∼ 300, the micelles maintain structural integrity, resisting shear-induced disturbances.

In addition, R_eff depends on both s and Pe. At low s and Pe ≤ 1, the clusters maintain N_c > 10 comparable to systems at Pe = 0, forming localized closure loops (Fig. 4(a) and (b); however, they exhibit constrained spatial organization. In medium s, the cluster adopts compact shear-independent configurations (N_c < 10, R_eff ≤ 5), characteristic of vesicles and micelles. As convective forces dominate over the magnetic interactions in the system (Pe > 1), the system stabilizes into small, well-defined structures (N_c ≤ 6, R_eff ≤ 5), irrespective of dipolar shift.

3.3 Shear effect on the structural properties of a dilute system of magnetic Janus particles at various dipolar shifts

As a summary of the observations presented in Fig. 2–4 we conducted an analysis to understand how shear and dipolar shifts affect the assembly of magnetic Janus particles. First, we initiated an analysis of the relative orientation between the particles. Fig. 5 shows the preferred orientation function (m_i·m_j)_max for varying s and Pe at λ = 45. When s results in a wide range of orientations, we calculated the mean orientation to quantify the predominant trend. This averaging process incorporates the error bars, as shown in Fig. 5, which reflect the variability in the orientation distribution rather than indicating a unique alignment.

Fig. 5 Effect of the shear in the particle orientations for values of dipolar shift at λ = 45 denoting three predominant orientational configurations. Panel shows the particle orientations of the maximum probability particle orientation as a function of s. The results for s = 0.6–0.8 were obtained through extrapolation of established trends in our dataset, showing consistent behavior with previous studies on similar systems. This notation is consistently used in all subsequent figures.

In Fig. 5, three aggregation modes are fundamentally governed by the relative dipole angle between neighboring particles, which dictates their assembly behavior. At low s and Pe ≤ 1, shear forces enhance the diffusion of clusters, and magnetic interactions tend to align the particles in a head-to-tail aggregation mode, favoring the formation of loops that align with the flow direction. However, as Pe increases in the medium s, convective forces begin to dominate the system, leading to the reorientation of the particles, which indicates a head-to-side aggregation mode, with some special cases in the antiparallel configuration. The extrapolated data shown in Fig. 5 agree with prior simulation studies of higher dipolar shifts (s ≤ 0.6), for which the preferred orientation is the antiparallel aggregation mode.^23,49

This second analysis was achieved by calculating 〈N_c〉 as a function of Pe, varying s (see Fig. 6a). The cluster growth dynamics reveal a dependence on both s and Pe. Compared to the quiescent system (Pe = 0), we observe an enhancement of 〈N_c〉 in all dipolar shifts, with different thresholds Pe. For low s, the enhancement of cluster growth is limited to low shear flow (Pe < 0.1 for s = 0, Pe ≤ 1 for s = 0.05–0.1), while medium s maintains growth until higher shear flow (Pe ≤ 1 for s = 0.2, Pe ≤ 5 for s = 0.3, and Pe ≤ 10 for s = 0.4–0.5). This behavior emerges from competing transport mechanisms: Brownian motion predominates over convection, and the displacement of particles is coupled with the flow field.⁶⁵ Particles in clusters experience a combination of rotational and elongational effects due to shear flow, which enhances their diffusive mobility. Similar behavior in micelles has been documented in studies by Bianchi⁴³ involving amphiphilic Janus particles.

Fig. 6 Shear effect on the cluster size at various dipolar shifts. Panel (a) shows the weighted averaged mean cluster size 〈N_c〉 under shear as a function of Pe at λ = 45. Panel (b) shows structural “phase” diagram. The graph highlights three zones: shear enhanced diffusion (yellow color, ), cluster breakage (green color, ), and no-cluster (pink color, ).

After reaching their enhanced growth cluster sizes, clusters transition into a dynamic equilibrium where fragmentation and rearrangement play significant roles. The repulsive barrier introduced by the applied shear forces competes with the magnetic interactions, resulting in more stable clusters that can persist over time (Pe ≤ 3 for low s, Pe = 20–45 for medium s).⁶⁶ Upon further increasing shear, clusters begin to fragment, as evidenced by the decrease in the average cluster size for all s. However, at Pe = 20, the system exhibits consistent behavior, with 〈N_c〉 stabilizing at 4 to 5 particles per cluster, representing micelles and quadruplets that remain unaffected by s. It is noteworthy that shear promotes the formation of fewer but larger aggregates, even at high Pe. Among all dipolar shifts, s = 0.5 stands out, maintaining more stable clusters up to Pe = 300, indicating resistance to shear-induced breakup. This behavior suggests that a higher lateral dipolar shift facilitates more efficient particle alignment and adaptability under strong shear forces, consistent with the enhanced stability observed in shifted dipole systems studied in previous research.^23,24,67 Moreover, this observation aligns with the findings of Moncho-Jordá et al.,⁴⁰ which indicated that cluster lifetimes are nearly independent of Pe for very low ϕ in hard sphere suspensions.

Fig. 6b presents a structural phase diagram based on the relationship between Pe and s, as shown in Fig. 6a. Three zones are delineated according to 〈N_c〉 (last 200 simulation steps): (i) shear-enhanced diffusion (yellow; SED), defined when 〈N_c〉 exceed the value in non-sheared systems (Pe = 0) at equivalents parameters 〈N_c〉_SED > 〈N_c〉_(Pe=0). (ii) No cluster (pink; NC): characterized 〈N_c〉 < 2 corresponding to predominantly single particles in simulations. (iii) Cluster breakage (green; CB): intermediate regime. The boundaries (dashed line) reflect physical transitions observed in simulations. The SED-CB threshold marks the onset of shear-dominated diffusion, while the boundary CB-NC (〈N_c〉 < 2) criterion captures the gas-dispersion state of cluster. Table 1 provides the approximate 〈N_c〉 < 2 and Pe ranges for each regime across different s values.

Table 1 Approximate thresholds of average cluster size 〈N_c〉 and Pe for shear-dependent regimes: SED, CB, and NC

Regime	Criterion	s	Pe range
Shear-enhanced diffusion (SED)	〈Nc〉SED > 〈Nc〉(Pe=0)	Low s	≲1
		Medium s	≲10
Cluster breakage (CB)	〈Nc〉SED > 〈Nc〉CB > 2
No cluster (NC)	〈Nc〉NC < 2	Low s	≳45
		Medium s	≳60

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In the SED zone, low Pe indicates that diffusion dominates over convection. The enhanced diffusion in this regime is attributed to a combination of rotational and elongational effects induced by shear flow, which promotes particle mobility and facilitates aggregation.⁴³ Consequently, the system forms clusters with higher 〈N_c〉 values and a more extended spatial organization, as shown in Fig. 4 and 5 compared to the quiescent system (〈N_c〉_(Pe=0)). The CB zone is characterized by shear forces that gradually dominate over magnetic forces, facilitating a rearrangement of clusters without leading to complete fragmentation. When the system transitions to higher values of Pe, it results in the fragmentation of the clusters into individual particles. This phase, characterized as the non-cluster state, exhibits gas-like behavior, with 〈N_c〉 reducing to two or fewer particles.

3.4 Combined effects of Pe and λ

The aggregation behavior of magnetic Janus particles under shear flow arises from competing mechanisms: stochastic Brownian motion that randomizes orientations, hydrodynamic shear that imposes directional alignment, and dipole–dipole interactions that promote particle aggregation. This three-way competition is well-documented in isotropic and conventional Janus systems.^39,46,49,68 In contrast to conventional Janus systems, the presence of a shifted magnetic dipole (s > 0) introduces an asymmetry that significantly modifies the torque equilibrium between particles.

To quantitatively assess this competition, we have adapted the torque-balance formalism established in prior studies of sheared Janus particles,⁶⁹ defining the critical ratio

where T^B is characterized by k_BT and T^H is given by 8πη[small gamma, Greek, dot above]a³/2, both of which favor cluster breakage. The restoring torque associated with interparticle interactions T^P has two contributions: (i) the magnetic dipole–dipole potential represented by μ₀|m²|/4πr_dij⁴ and (ii) the magnetic torque associated with dipole displacement leading to sμ₀|m²|/4πr_dij⁴. In the case of two contacting particles with low s, the minimum distance between dipoles is r_dij ∼ 2a. Due to the lateral displacement of the dipole for each particle, the dipole–dipole separation r_dij becomes smaller than the particle diameter. However, for the sake of analytical tractability and consistency in scaling arguments, we introduced the scaling r_dij ∼ 2a in the simulation. As a result, T^P scales as λ(1 − s). This yields the following dimensionless equation:

The R_T quantifies the balance between cluster aggregation and a non-clustering state in our magnetic Janus system. When R_T ≪ 1, restorative torques predominate, facilitating stable cluster formation as dipolar alignment and strength prevail over Brownian randomness and shear-induced fragmentation. Conversely, at R_T ≫ 1, the combined disaggregation torques compromise the magnetic interactions, resulting in the fragmentation of the cluster.

The theoretical framework established through this simple torque balance analysis finds direct validation in the phase diagram presented in Fig. 7, which illustrates the overall structural changes of magnetic Janus particles under the influence of shear flow, for different values of s ≤ 0.5, λ ranging from 15 to 60, and Pe spanning from 0 to 100. Cluster structures were identified using orientational distribution functions and N_c during the stationary state (as outlined in Fig. 4). The red continued line in Fig. 7 corresponds to the threshold R_T ∼ 1, which precisely separates the aggregation state (R_T < 1) from the non-clustering state (R_T ≥ 1), indicated in pink. Similarly, the red dashed line in panel f demarcates the transition between the aggregation and non-clustering states; however, here, the critical R_T differs from unity, as analyzed in the S2 (ESI†).

Fig. 7 Structural “phase” diagram of cluster phases observed at steady state varying λ and Pe, for dipolar shifts (a) s = 0.0, (b) s = 0.1, (c) s = 0.2, (d) s = 0.3, (e) s = 0.4, and (f) s = 0.5. The red line marks the theoretical R_T separating aggregation state and no clustering (pink color). Symbol colors denote cluster types: (loops and rings); + (chains); (micelles); (vesicles and amorphous rings); (quadruplets); (tetrahedrons, triplets, and doublets), and (gas-like structure).

At low s, loops predominate regardless of λ value (see Fig. 7(a) and (b)). These loops undergo progressive size reduction with increasing Pe, following a well-defined transition sequence: elongated loops, fragmented small chains, isolated triplets and doublets, and ultimately gas-like structures, with a preferred head-to-tail orientation. In medium s, the magnetic restoring torque plays a critical role in stabilizing characteristic vesicle and micelle structures with predominant head-to-side and antiparallel configurations (see Fig. 7(c)–(f)). As convection dominates, the system undergoes gradual structural fragmentation as a consequence of lower shifts and exhibits distinct resilience under stronger magnetic interactions (λ = 45–60) and even higher Pe (Fig. 7f). Furthermore, for high dipolar shifts (s = 0.6–0.7), we extrapolated the phase behavior based on empirical trends from our data and prior studies. This analysis reveals a breakdown of the theoretical R_t boundary that separates aggregated from non-clustering states. Instead, the system transitions into a regime dominated by micellar phases, tetrahedral arrangements, and localized triplet/doublet configurations, consistent with the strongest directional dipole–dipole interactions at s values near 1. These structures suggest that robust magnetic coupling can counteract shear-induced fragmentation, thus maintaining cluster integrity even under substantial lateral shifts. This behavior underscores the critical role of magnetic strength in determining the stability and assembly of magnetic Janus clusters, consistent with findings in paramagnetic particle systems subjected to shear flow.⁴⁹

4 Conclusions

We have employed Brownian dynamics (BD) simulations to investigate the shear-driven assembly and breakup of dilute suspensions of magnetic Janus particles with laterally shifted dipoles. Our findings underscore the critical role of the dipolar shift, s, and the Péclet number, Pe, in governing the aggregated structures.

At low dipolar shifts (s < 0.2) and low Pe, Brownian motion predominates, resulting in rapid aggregation into elongated chains and loop-like clusters. Shear flow within this regime enhances particle encounters by promoting rotational and elongational effects, thereby accelerating the increase in cluster size. However, with an increase in Pe, shear forces progressively disrupt these clusters, fragmenting them into smaller, more stable aggregates or even dispersing them into gas-like phases. In contrast, larger dipolar shifts (s > 0.3) lead to the formation of compact, vesicle- and micelle-like assemblies, whose enhanced stability is maintained at higher Pe. We attribute this resilience to the significant lateral displacement of the dipole, which supports robust magnetic coupling and facilitates orientations, particularly head-to-side or antiparallel, that can withstand substantial shear stresses.

Orientational analyses reveal that dipoles in low-s systems tend to align in parallel configurations, favoring loop structures and chain-like aggregates. Conversely, medium and high dipolar shifts result in more complex dipole orientations under shear, demonstrating that lateral dipole displacement can significantly alter local bonding motifs and cluster morphologies. These observations highlight that both the magnitude of the dipolar shift and the relative strength of the shear flow are essential for tuning the interplay between magnetic attraction and hydrodynamic disruption.

The competition between structure-breaking and structure-restoring torques described by R_T ultimately determines whether particles remain aggregated or disperse under shear. When R_T ≪ 1 restorative dipolar forces dominate, enabling the formation of loops, chains, or micellar clusters whose morphology depends on the s. Conversely, at R_T ≫ 1 shear and thermal effects overcome magnetic interactions, leading to fragmentation into smaller aggregates or fully dispersed states. The threshold R_T ∼ 1 marks a sharp crossover between these regimes. The inclusion of the dipolar shift in R_T scaling reveals how lateral dipole displacement enhances cluster stability—a key distinction from conventional Janus systems.

Taken together, these results have significant implications for the design and manipulation of anisotropic colloidal systems in flow environments. Materials incorporating magnetic Janus particles with tunable dipolar interactions can achieve tailored structural integrity under diverse shear conditions, enabling novel applications in areas such as magnetorheological fluids, where stable yet switchable clustering is often desirable, as well as in microfluidic devices and drug delivery platforms, where controlled aggregation and dispersion are essential for targeted transport and release. By optimizing s and the dipolar coupling constant λ, researchers and engineers can exert precise control over particle stability, dynamic reconfiguration, and the resultant macroscopic properties. Future work may extend these simulations to higher particle concentrations, examine the influence of external magnetic fields in conjunction with shear, or incorporate hydrodynamic interactions to enhance our understanding of these complex nonequilibrium colloidal microstructures.

Conflicts of interest

There are no conflicts of interest to disclose.

Data availability

The code and datasets supporting this study are archived in Zenodo DOI: https://doi.org/10.5281/zenodo.15338102. Supplementary videos and figures documenting the dynamic aggregation processes are included in the ESI.†

Acknowledgements

This research was supported by the U.S. National Science Foundation's Wisconsin-Puerto Rico Partnership for Research and Education in Materials, Funder ID No. 10.13039/100000001, Award No. 1827894.

Notes and references

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Footnote

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d5sm00457h

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