Self-assembly of magnetic Janus colloids with radially shifted dipoles under an external magnetic field

Abstract

Magnetic Janus particles (MJPs) with radially shifted dipoles exhibit a versatile platform for engineering responsive materials through field-directed self-assembly. Motivated by their potential in programmable soft matter, Brownian dynamics simulations are used to systematically investigate how the radial dipolar displacement s and the Langevin parameter α govern the aggregation pathways and emergent morphologies of MJPs in quasi-two-dimensional environments. We identified six distinct aggregation regimes: three arising under low magnetic fields (α ≲ 10) corresponding to the low-, intermediate-, and high-shift cases, and two emerging at intermediate (10 ≲ α ≲ 90) and high magnetic fields (α ≳ 90). These regimes exhibit a rich morphological evolution as α increases: from disordered loops (low α, low s), islands (low α, intermediate s), and worm-like clusters (low α, high s), transitioning through chiral and tangled chains (intermediate α, intermediate and high s), and culminating in fully aligned chains (intermediate α with low s, and high α for all s). A structure diagram predicted by considering a simple ratio of competing torques (R_Mag) effectively illustrates these transitions and specifies the conditions necessary for structural reorganization. This framework supports the rational design of adaptive colloidal architectures for applications in targeted delivery, soft microrobotics, and reconfigurable magnetic systems. Notably, the universal convergence to a growth exponent of z ≈ 0.473 under high magnetic fields (α ≳ 90) reveals a definitive kinetic signature of complete cluster alignment along the field direction, establishing a robust and tunable route to field-induced material organization.

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1 Introduction

Magnetic colloids have gained prominence in biomedical and materials science disciplines due to their remotely controllable behavior under applied magnetic fields.^1–3 In biomedicine, they serve as contrast agents in magnetic resonance imaging (MRI) and as localized heat sources in magnetic hyperthermia treatments. Their appeal lies in the tunability of magnetic interactions, which, unlike electrostatic forces, are largely insensitive to ionic strength, thus enabling consistent manipulation under physiological conditions.^1,3 Beyond biomedical settings, magnetic colloids act as field-responsive building blocks in reconfigurable systems such as photonic crystals, microfluidic actuators, and soft robotic materials.^4,5

In recent decades, Janus particles—colloids characterized by a break in surface symmetry—have emerged as versatile materials due to their directional interactions and anisotropic assembly behavior.^6,7 Magnetic Janus particles (MJPs), which integrate both magnetic responsiveness and surface anisotropy, thus represent a hybrid platform for creating multifunctional field-directed architectures.⁸ Their duality offers a unique opportunity to engineer self-assembled systems with adaptive properties governed by external stimuli.

Two idealized models are commonly used to describe MJPs:^9–11 (i) a radially shifted dipole, in which the magnetic moment is displaced along the particle radius and remains collinear with it; and (ii) a laterally shifted dipole, in which the magnetic moment is displaced tangentially with respect to the particle center. The two geometries break symmetry in distinct ways and thus lead to different torque balances and assembly pathways. This study focuses exclusively on MJPs with radially shifted dipoles subjected to a uniform external magnetic field.

A wide range of experimental and computational studies has uncovered diverse structural motifs in magnetic Janus systems, including head-to-tail chains, closed loops, stacked columns, helical configurations, and chiral assemblies, all emerging from variations in particle anisotropy and external driving conditions.^9,10,12,13 Yet, a unified framework linking radial dipole displacement to field-induced reorganization thresholds and kinetic signatures remains elusive. Addressing this gap is particularly relevant, as radially shifted dipoles naturally arise in Co/Pt multilayer depositions and patterned coatings, where the magnetic moment is typically displaced by 65–75% of the particle radius.^14–16 This intrinsic feature enables direct experimental validation of model predictions. These fabrication methods ensure that the magnetic dipole remains oriented perpendicular to the particle–fluid interface, consistent with a radially shifted configuration. Alternative strategies employing embedded inclusions offer additional control over dipole positioning but are often constrained by technical complexity and limited reproducibility.¹⁷ As a result, systematic investigations into how the degree of dipolar displacement influences field-directed assembly have remained largely theoretical.^9,13

Computational investigations have revealed that MJPs with radially shifted dipoles exhibit diverse assembly behavior even in the absence of applied fields, forming curved or chiral motifs due to asymmetric magnetic interactions.^9,13 Studies incorporating external fields, though limited to laterally shifted dipoles, have reported structures such as helical chains and stacked columns.^10,12 However, the dynamic response and equilibrium morphology of MJPs with radial dipole shifts under uniform external fields remain unexplored. This represents a critical gap in our understanding of how internal and external torques compete to determine the organization of the colloids.

In this work, we investigate the field-driven self-assembly of MJPs with radially shifted magnetic moments using Brownian dynamics simulations in a quasi-two-dimensional (q2D) setting. A modified dipolar hard-sphere (DHS) model is employed that accounts for a fixed radial displacement of the magnetic moment from the particle center toward its surface, along a direction collinear with the dipole orientation.^18–20 This configuration is consistent with experimentally realizable Janus geometries and ensures that the dipole remains perpendicular to the magnetized surface region.^9,13,17 The system is subjected to a uniform, static magnetic field applied in the plane of motion, facilitating investigation of competition between field alignment and interparticle interactions. This analysis produces a compact structure diagram and a simple, predictive torque ratio R_Mag that rationalizes reconfiguration thresholds. Our results expose six aggregation regimes and a universal kinetic convergence under strong fields, offering a design rule for field-programmable soft architectures.

The simulations are conducted within the overdamped Brownian dynamics framework, which incorporates stochastic translational and rotational diffusion, magnetic torques, and steric repulsion, while neglecting hydrodynamic interactions, an acceptable approximation under dilute conditions.^21–23 The external field imposes a Zeeman torque on each particle, competing with dipole–dipole interactions that may favor alternative local arrangements.²⁴ The resulting morphologies emerge from this balance between thermal fluctuations, field-induced alignment, and anisotropic interparticle forces.

The structure of this paper is delineated as follows. Section 2 presents the model alongside the governing equations that elucidate the magnetic dipole–dipole and dipole–field interactions, thereby establishing the foundation for understanding the evolution of the colloidal system. Section 3 analyzes the impact of field intensity on aggregate morphology and elucidates the correlation between dipolar displacement and particle orientation. Finally, Section 4 summarizes the main findings of this research endeavor.

2 Problem formulation

2.1 Model system

The system under consideration is a quasi-two-dimensional colloidal suspension, comprising N identical ferromagnetic spherical particles, which are dispersed within a Newtonian fluid characterized by a viscosity of η and a density of ρ, at a specific surface fraction denoted as ϕ_s. Each particle is modeled as a dipolar hard sphere (DHS); refer to Fig. 1, characterized by a fixed magnetic dipole moment m situated at an off-centered position within the particle. This is described by the dimensionless dipolar shift, which is the ratio of the radial displacement of the dipole S to the radius of the particle a, s = S/a. The parameter s extends from 0 (indicating a dipole placed at the geometric center) to 1 (indicating a dipole positioned at the surface of the particle).

Fig. 1 Model system, consists of spherical colloidal particles of radius a, where the diagram highlights the magnetic region of each particle in orange. The magnetic moments (m_i and m_j) are radially displaced from the center of mass of the particle by a radial distance S. r_ij and r_dij are the distances between the center of mass and the magnetic dipoles of the particles i and j. We define the simulation domain using a fixed inertial reference frame (x, y, z), while each colloid possesses its own local, mobile coordinate system (x′, y′, z′). A uniform external magnetic field with intensity H is applied along the positive y-direction.

The presence of a permanent magnetic dipole m allows each particle to interact magnetically with others through the classical dipole–dipole interaction potential. This potential governs both short- and long-range interactions and is responsible for the self-assembly of the particles into clusters. Furthermore, when an external magnetic field is applied, the Zeeman interaction²⁴ introduces a directional torque that competes with the dipolar interaction, thus influencing both the orientation and morphology of the aggregates. Due to their colloidal scale, particles are subjected to significant Brownian motion arising from thermal fluctuations of the surrounding fluid. In this regime, inertial effects are negligible.

Fig. 1 also illustrates the model system with the inertial frame of reference (x, y, z) fixed to the system. Furthermore, each particle has its own mobile reference frame (x′, y′, z′). The positions and orientations of each particle are initialized randomly.

We employ a q2D geometry to (i) mirror common experimental conditions where particles are confined near interfaces or shallow microfluidic chambers; (ii) isolate the competition of torques by suppressing out-of-plane reorientation modes that obscure in-plane chaining and chiral locking; and (iii) improve statistical sampling of long chains at fixed computational cost. This choice targets the regime most relevant for field-guided prototyping and imaging, while preserving the essential dipolar and Zeeman physics under dilute conditions.

2.2 Brownian dynamics simulations

The described colloidal system provides a simplified yet representative physical model for magnetic Janus particles that exhibit anisotropic interactions. To investigate its dynamic behavior, we employ Brownian dynamics (BD) simulations, this method is particularly suitable for modeling overdamped systems at the colloidal scale and effectively captures both deterministic and stochastic contributions to particle motion, including translational and rotational diffusion.²⁵ These stochastic components are essential for enabling collisions and reconfiguration events that ultimately drive the aggregation process.

BD incorporates three principal types of interactions: Brownian motion induced by thermal fluctuations in the surrounding Newtonian fluid, hydrodynamic drag forces, and interparticle interactions. Due to the colloidal length scale, inertial effects are considered negligible and system dynamics is assumed to be instantaneous. The particles are confined within a square simulation box with periodic boundary conditions, and the total number of particles N remains constant throughout the simulation. Under these assumptions, the particle dynamics, both translational and rotational, is governed by the overdamped Langevin equations:²⁶

F^B + F^D + F^P = 0, (1)

T^B + T^D + T^P + T^H = 0. (2)

In this framework, F^B and T^B represent the Brownian force and torque, while F^D and T^D denote the drag-induced force and torque. The terms F^P and T^P correspond to the cumulative forces and torques resulting from the interactions between particles. Furthermore, T^H accounts for the torque exerted by the external magnetic field. The model characterizes interparticle interactions through both a surface potential and a magnetic potential. The magnetic potential (Φ(r_dij)) is described by the following expression:²⁷

r_dij = r_di − r_dj, r_dij = ‖r_dij‖.

Here, μ₀ denotes the vacuum permeability and r_dij represents the distance between the dipoles. The surface potential contributes solely to a repulsive force confined to a steric shell of thickness δ surrounding each spherical colloid. This repulsion is modeled through a steric force expression that characterizes surface interactions:^28,29
where k denotes Boltzmann's constant, T is the absolute temperature of the fluid, λ_R represents the steric layer coefficient, and r_ij corresponds to the center-to-center distance between two interacting particles. The force and torque between particles are ultimately characterized by the following equations:

F^P_ij = −∇(m_i·B_ij) + F^R_ij, (5)

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

T^H = μ₀mHn_i × h, (7)

where B_ij denotes the magnetic field produced by the dipole m_j acting on the dipole m_i (Φ(r_dij) = m_i·B_ij), m is the magnitude of the magnetic dipole m, H is the magnitude of the externally applied magnetic field H oriented along the direction h. This field remains constant throughout the simulation, maintaining both its magnitude and direction.

The positions and orientations of the particles are determined by numerically integrating the overdamped Langevin eqn (1) and (2). A finite-difference scheme is employed to perform this integration, resulting in the following dimensionless expressions:

where in these expressions, I denotes the identity tensor, while Δr̃^B and Δ[capital Omega, Greek, tilde]^B represent the dimensionless Brownian translational and rotational displacements, respectively. The temporal variable is scaled using the translational Brownian time scale (t ∼ τ_D), whereas length, force, and torque are nondimensionalized by the particle radius (r ∼ a), the characteristic Brownian force (F ∼ kT/a), and the Brownian torque (T ∼ kT), respectively.

The behavior of MJPs is characterized by the dipolar shift s and two dimensionless groups: the dipolar coupling constant λ, given by

which quantifies the relative strength of magnetic interparticle interactions compared to thermal (Brownian) forces, and the Langevin parameter α, defined aswhich gauges the relative impact of magnetic field intensity on the Brownian motion of the particles. Together, λ and α, determine the primary balances between dipolar interactions, field alignment, and Brownian motion.

The Brownian contributions to displacement and rotation in eqn (8) and (9) are expressed in terms of the diffusion tensors as follows:

〈Δr̃^B_i〉 = 0, 〈Δr̃^B_iΔr̃^B_i〉 = 2IΔt̃, (12)

The rotation of a particle in three dimensions presents challenges in tracking; therefore, quaternion parameters are utilized to achieve correct orientation.³⁰ Quaternion parameters can be defined in terms of Euler angles.³¹

In the present q2D setup, translation occurs in the (x, y) plane while rotations are fully three-dimensional via quaternions. This retains the correct dipole reorientation dynamics while constraining particle centers to a single plane, consistent with interface-confined experiments.

2.3 Cluster properties

The identification of particles within a cluster is based on a bounding factor, a dimensionless criterion that evaluates the proximity between particles.³² Specifically, two particles are considered to belong to the same cluster when the ratio of their center-to-center distance to the radius of the particle is 1.2.^9,33,34

Starting from a randomly distributed configuration, the temporal evolution of cluster formation is governed by deterministic particle–particle interactions (i.e., magnetic and steric repulsion), particle-medium interactions (e.g., hydrodynamic drag and magnetic field effects), and thermal fluctuations represented by Brownian motion. In this context, the nucleation factor (n_c) serves as a key metric to monitor the progression of the system from a dispersed state, where the number of clusters (N_c) is approximately equal to the number of singlets (N_s) to increasingly larger clusters. The nucleation factor is defined by

It is important to emphasize that the simulation is performed under periodic boundary conditions, ensuring that N remains constant throughout the simulation.

Another critical parameter for characterizing the aggregation is the size of the cluster and its evolution over time. To this end, we employ the weighted average mean cluster size, denoted as 〈N_c〉, which provides a statistically representative measure of the system by averaging cluster sizes weighted by the number of particles per cluster (N_c,p) and the total number of clusters (N_c) according to the following expression:

Finally, the orientation distribution function is a critical metric utilized to characterize the internal microstructure of particle clusters, providing information on the relative alignment of neighboring magnetic dipoles by quantifying the frequency of dipole orientations between adjacent particles, it is defined as P = P((m_i·m_j)/‖m‖²), where (m_i·m_j) is the dot product between the dipole moments of two neighboring particles and ‖m‖² their squared magnitude, versus their relative orientation, defined by (m_i·m_j)/‖m‖².

2.4 Simulation parameters

Previous studies have shown that the dipolar shift s exerts a significant influence on the final structural configuration of the clusters.⁹ The range of values of s considered in this work ranges from 0.0 to 0.75. In the case of s ≳ 0.75, the close proximity of the magnetic dipoles between particles markedly amplifies their magnetic interactions but simultaneously makes the simulation numerically challenging. Notably, this range aligns with the practical upper limit of dipolar shifts realized through Co/Pt multilayer deposition and patterned coating methodologies.^14–16 Furthermore, the extrapolation to s > 0.75 will be discussed in detail in Section 3.5, which supports a consistent morphological transition beyond s ≈ 0.75. This reinforces the predictive capability of our framework even in the absence of direct simulations.

To examine the influence of the magnetic field on dynamic aggregation, we consider the values of α, which serves as a measure of the strength of the applied external magnetic field, within the range from 0 to 90. This spectrum extends from weak to strong field regimes, thereby encompassing conditions pertinent to both passive (thermally dominated) and active (field dominated) alignment scenarios. The dipolar coupling constant is maintained at λ = 45, corresponding to a regime in which stable aggregation occurs without immediate saturation, as delineated in the prior structure diagram proposed by Victoria-Camacho et al. (2020).⁹ At this specified value, a distinct differentiation among the three aggregation modes is observed, while the magnetic interactions remain sufficiently strong to facilitate the formation of stable clusters without precipitating excessive aggregation.

Simulations are performed under dilute conditions, where hydrodynamic interactions are negligible,^22,23,35 thereby isolating torque-driven dynamics. A surface fraction of ϕ_s = 0.01 is chosen to reproduce these conditions and emphasize dipolar interactions.^35–39 Under these settings, the results are insensitive to the dimensions of the simulation box and to the integration time step. Our main analysis focuses on ϕ_s = 0.01. Increasing ϕ_s enhances the collision frequency, promotes inter-chain contacts, and broadens the regime of tangled aggregates; at sufficiently high ϕ_s, the system may evolve toward percolated networks or dipolar gels. A comprehensive concentration-dependent study will be pursued in future work.

To ensure statistical robustness, simulations were performed with N = 500 particles and averaged over five independent realizations. The absence of finite-size artifacts was verified by comparing systems with N = 500 and N = 1000 at fixed ϕ_s, yielding indistinguishable morphologies, orientation distributions, and growth exponents within statistical uncertainty (see Fig. S1 in the SI). The agreement across system sizes confirms that N = 500 is sufficient to capture the intrinsic structural transitions governed by dipolar and field interactions.

Typical colloidal MJPs span radii of a ≃ 0.25–2.5 µm. For fixed material and temperature, the dipolar coupling parameter λ decreases with increasing particle size a, whereas the field strength parameter α increases linearly with H. Consequently, similar structural regimes can be accessed by co-tuning the particle size, magnetic moment m, and applied field H. The q2D geometry adopted here mirrors interface-confined assemblies commonly employed in experimental imaging and actuation studies.

3 Results

3.1 Morphological evolution

To explore the dynamic evolution of the cluster morphology in MJPs, we analyze simulation snapshots for various s and α. The evolution of morphology can be interpreted as a torque balance process between two dominant mechanisms: (i) the alignment torque imposed by the external field T^H, and (ii) the dipolar reconstructive torque T^P, which attempts to stabilize energetically favorable cluster configurations. The resulting structures are determined by the relative magnitudes of T^H and T^P, which dictates whether the chains align, twist, or resist reconfiguration.

The effect of α on the structure of the clusters is particularly notable in long-term simulations. Based on previous work by Victoria-Camacho et al. (2020),⁹ three characteristic aggregation regimes are identified according to the value of s: low-shift regime (s ≲ 0.35), intermedaite-shift regime (0.35 ≲ s ≲ 0.65), and high-shift regime (s ≲ 0.65). As α increases, it perturbs the dynamic evolution and induces a significant reorganization of the structures, as shown in Fig. 2. In the low-shift regime, even weak magnetic fields reorient the clusters along the field direction, suppressing the formation of loops and promoting single-chain configurations (Fig. 2e and f). These structures are particularly sensitive because of their linear dipole alignment, making them highly susceptible to magnetic torque.

Fig. 2 Dynamic evolution of cluster morphology for different dipolar shifts s and field intensities α under a high-λ regime (λ = 45). The figure shows representative snapshots at increasing times for each system. Three aggregation modes are identified: s = 0.3 (low shift), s = 0.5 (intermediate shift), and s = 0.7 (high shift). As α increases, clusters reorganize into field-aligned structures, with varying degrees of resistance. High-shift systems form chiral chains that persist until strong magnetic fields induce full alignment.

In the intermediate-shift regime (Fig. 2g and k), the clusters exhibit moderate resistance to alignment. At low α, chains form with misaligned segments that do not strictly adhere to the field direction. However, with increasing field strength, this resistance is gradually overcome, resulting in alignment similar to that observed in the low-shift regime. The intermediate-shift regime represents a transition state in which neither torque dominates entirely. This leads to complex configurations such as misaligned or staggered chains, where both steric accessibility and dipolar frustration play crucial roles. The competition results in partial alignment that reflects the nuanced balance between particle-level interactions and the global orientational bias imposed by the field.

Finally, in the high-shift regime (Fig. 2h, l, p and t), clusters are stabilized by a central magnetic core, with minimal dipolar interaction at the edges. These clusters exhibit strong internal cohesion and resist reorientation. At intermediate field strengths, the particles begin to form chiral chains, ladder-like structures characterized by intercalated arrangements with relative angles that deviate from collinearity. This resistance weakens only at high α values, where chiral chains eventually collapse into simple aligned chains (Fig. 2x). These observations demonstrate that the morphology of clusters is closely governed by both dipolar interactions and applied field strength. Low- and intermediate-shift regimes offer minimal resistance to field-induced reorganization, while high-shift systems exhibit quantifiable and progressive alignment thresholds.

These regimes can be interpreted as structural responses to energetic landscapes shaped by the interplay between the dipole shift s and the field strength α. In the low-shift regime, the external alignment torque T^H easily overcomes the minimal internal resistance to reorientation, allowing the particles to minimize their dipolar potential energy by forming linear chains aligned with the field. In contrast, at higher s, the internal dipolar torques T^P are geometrically amplified due to the increased lever arm associated with the shifted dipole. As a result, proportionally larger values of α are required to overcome these internal torques and achieve alignment. The corresponding energy landscapes exhibit local minima associated with twisted or antiparallel configurations, leading to delayed or frustrated reorientation unless the field-induced torque becomes sufficiently dominant. This morphological diversity highlights the critical role of torque asymmetry in dictating self-assembly pathways in MJPs.

3.2 Aggregation dynamics

The dynamic aggregation process in MJPs can be divided into two main stages: the formation of primary structural units (nucleation) and the subsequent growth of larger clusters. During nucleation, isolated particles (singlets) interact through magnetic and steric forces to form basic configurations such as doublets, triplets, or quadruplets. These configurations serve as building blocks for subsequent aggregation. This two-stage process, nucleation and growth, is quantitatively characterized using the nucleation factor n_c (eqn (14)), which tracks the transition from dispersed particles to organized clusters.

From a geometric and dynamic perspective, dipolar displacement s plays a pivotal role in the early stages of aggregation. At low s, symmetric dipolar fields enable straightforward singlet pairing via head-to-tail alignment, and strong head-to-tail coupling facilitates fast nucleation. In contrast, at higher s the angular constraints and steric barriers are introduced due to off-center dipole positioning, increasing the configurational energy barriers. This results in steric frustration and angular offsets that delay nucleation unless a sufficiently strong magnetic alignment torque is present. This behavior reflects a balance between thermal motion, dipolar potential wells, and torque imparted by the external field. As s increases, the anisotropic magnetic field of each particle generates multiple local minima for alignment, increasing configurational entropy and requiring stronger fields to guide the system toward ordered growth.

As shown in Fig. 3a–d, the magnetic field strength α does not significantly influence nucleation in low-shift systems. However, in the intermediate-shift regime (Fig. 3c), the growth process accelerates in the presence of a magnetic field, occurring earlier than in its absence. In this case, the magnitude of α (low or high) does not alter the qualitative trend: any non-zero magnetic field induces earlier aggregation. In contrast, the high-shift regime (Fig. 3d) exhibits enhanced sensitivity to the magnetic field, with both nucleation and growth increasingly driven by α.

Fig. 3 Nucleation and growth dynamics as a function of dipolar shift s and magnetic field intensity α. (a)–(d) Evolution of the nucleation factor n_cversus dimensionless time t/τ_D for different values of s. (e)–(h) Weighted average cluster size 〈N_c〉 versus dimensionless time t/τ_D for different values of s. A power-law scaling 〈N_c〉 ∼ t^z is observed in the growth phase. Field intensity modifies the onset and speed of aggregation, with all systems converging to a universal growth rate of z ≈ 0.473 under strong fields.

The rate of aggregation is further analyzed using the average size of the cluster 〈N_c〉 (eqn (15)), which exhibits a power-law time dependence, 〈N_c〉 ∼ t^z (Fig. 3e–h). This behavior implies a non-saturated unbounded cluster growth. For s = 0.0, the exponent z changes only slightly with α, ranging from z ≈ 0.477 at α = 0 to z ≈ 0.473 at α = 90, and shows a progressive increase for s = 0.3. These values converge to z ≈ 0.473 for all systems under strong fields, confirming a universal aggregation mode dominated by field-aligned chains. In the intermediate-shift regime (Fig. 3g), the aggregation rate shows a qualitative jump from z ≈ 0.356 (bidirectional growth at α = 0) to z ≈ 0.459 at α = 10, reaching z ≈ 0.473 at α = 90. This indicates that moderate field intensities can transform bidirectional island-like growth into linear chains. For high-shift systems (Fig. 3h), the rate increases more gradually (e.g., from z ≈ 0.375 to z ≈ 0.473), reflecting persistent resistance to alignment that is only overcome at high field strengths.

In general, the saturation value z ≈ 0.473 appears to represent a universal limit under strong magnetic fields. Regardless of the dipolar shift, strong external fields lead to the same morphological outcome: the formation of field-aligned chains. This behavior suggests that, under such conditions, the aggregation dynamics of all magnetic colloids, Janus or otherwise, tend to converge toward a common trajectory.

The exponent z can be interpreted as a signature of the dominant growth pathway. The convergence z ≈ 0.473 under high-field intensities (α ≳ 90) indicates a fast one-dimensional chain assembly. This behavior is reminiscent of the universal scaling observed in classical models of diffusion-limited cluster aggregation (DLCA),⁴⁰ where fractal-like structures form through the irreversible aggregation of diffusing particles. In such systems, power-law scaling in cluster size growth often signifies the dominance of self-similar kinetic pathways governed by underlying symmetry-breaking mechanisms or dimensional constraints. This provides a direct link between the observable aggregation dynamics and the underlying symmetry of the interaction potential.

The observed growth exponent z ≈ 0.473 under strong magnetic fields resembles ballistic aggregation behavior, characterized by unidirectional and irreversible particle aggregation. This contrasts with DLCA (z ≈ 0.33) and reaction-limited cluster aggregation (RLCA), where z ≈ 0.5 but with significant polydispersity and branching.⁴⁰ The convergence to z ≈ 0.473 indicates a unique kinetic regime that combines directional field bias with Brownian motion, leading to quasi-ballistic linear growth. This universal behavior has not previously been reported in Janus systems and marks a distinct transition to one-dimensional chain growth. This convergence results from the collapse of structural degrees of freedom under strong directional driving. When magnetic torque dominates over dipolar forces, the system is effectively projected onto a one-dimensional configurational subspace with a single energetically favorable orientation along the field. This dimensional reduction underlies the observed kinetic universality.

In MJPs, this field-aligned regime induces directional anisotropy that reduces the effective dimensionality of the aggregation, favoring uniaxial chain formation. The suppression of branching and loop formation under strong fields collapses the configurational phase space, enforcing a universal trajectory regardless of microstructural diversity or dipolar shift. The value z ≈ 0.473 thus emerges as a kinetic exponent characterizing the one-dimensional ballistic-like growth of chains under steady-field alignment. The magnetic field acts as a directional bias that gradually overcomes angular frustration. Although low-shift systems lack significant internal torque resistance, high-shift systems exhibit restoring torques from misaligned dipoles. This explains the delayed but abrupt onset of ordered growth in the high-shift regime once the field is applied.

Such universal behavior also aligns with findings in the literature on percolation-limited aggregation and self-affine growth in systems subjected to directional driving forces. It indicates a kinetic fixed point, where the field-dominated aggregation becomes independent of the underlying dipole anisotropy. This suggests that for applications requiring predictable and linear aggregation (e.g., in tunable soft actuators or magnetically driven microrobotic arms), adjusting the field intensity alone may be sufficient to mitigate the complexities introduced by intrinsic particle asymmetries.

This mechanism is analogous to constrained motion in systems where external fields suppress thermal branching, leading to strongly anisotropic growth fronts. The suppression of angular degrees of freedom by the external field not only controls directionality but also limits the emergence of competing structural motifs, thereby simplifying the aggregation landscape.

3.3 Field-induced reorganization

To quantify how the cluster morphology is affected by the magnetic field, the orientation distribution function P(m_i·m_j) as a function of (m_i·m_j)/‖m‖² is used, where (m_i·m_j) denotes the dot product of the magnetic dipole moments of two neighboring particles, and ‖m‖² is the squared magnitude of the dipole moment. This function measures the frequency of relative orientations between neighboring dipoles, providing insights into the internal structure of the clusters. The orientation distribution function encapsulates the balance of torques acting on dipoles within clusters. A narrow distribution centered on 0° indicates the dominance of the alignment torques, whereas a broad or multimodal distribution reflects competing influences, such as steric hindrance or multiparticle frustration. The shape and spread of P(m_i·m_j) thus encode the local mechanical stability of the structures. Morphological identity is not determined solely by a unique orientation, but rather by the distribution of possible alignments. A visual summary of this behavior is presented using box plots (Fig. 4a–c), which capture the spread and median of relative angles for different values of s and α, the variation in these distributions reflects the presence of competing energy minima in the landscape of dipolar interaction.

Fig. 4 Orientation distribution function analysis across shifted dipole regimes. (a)–(c) Box plots of the relative orientation (m_i·m_j)/‖m‖² for neighboring particles under varying s and α. (d)–(i) Orientation distribution function P = P((m_i·m_j)/‖m‖²) for s = 0.0 to s = 0.75 as a function of Langevin parameter α. (j)–(o) Structural reorganization in the presence of weak to strong magnetic fields, leading to suppression of loops and emergence of chiral chains and aligned chains. The final configurations at high α, on all systems, exhibit fully aligned chains.

In the low-shift regime, the absence of a magnetic field (α = 0) facilitates the formation of head-to-tail chains. These structures typically align at 0°, but the presence of loops and small rings introduces a distribution of angles that can extend up to 60° (Fig. 4d and e). However, once a magnetic field is applied, the system undergoes a transition to exclusively aligned chains at 0°, effectively suppressing loop formation (Fig. 4j and k). In this regime, the symmetric position of the dipole ensures that the torque vector aligns with the particle axis, enabling efficient alignment and propagation of head-to-tail chains. The suppression of loops under the field is driven by the minimization of both magnetic potential energy and rotational entropy, yielding morphologies optimized for linear propagation.

In the intermediate-shift regime, the structural diversity becomes more pronounced. In the absence of a magnetic field, the clusters exhibit a wide distribution centered around 60° to 120°, indicative of bidirectional or island-like morphologies (Fig. 4f and g). When a magnetic field is applied, for s ≲ 0.5 the orientation distribution shifts toward 0° forming field-aligned chains, for s ≳ 0.5 the orientation distribution shifts toward 0° to 90° forming intercalated chains that resist complete alignment (chiral and tangled chains). At high-field intensities (α ≳ 90) this resistance decreases, resulting in the emergence of field-aligned chains (Fig. 4l and m). The structural diversity observed in the intermediate-shift regime arises from the interplay between dipole orientation, edge asymmetry, and steric packing constraints. Chiral chains emerge as metastable compromises: partially aligned structures that retain internal angular coherence while responding to the external torque. The gradual suppression of the angular dispersion with increasing α reflects the realignment torque that progressively overcomes these internal constraints.

In the high-shift regime, orientation diversity is particularly pronounced, the close proximity of the magnetic moment to the particle edge increases both the torque magnitude and the susceptibility to orientation frustration. This introduces local metastable states that delay the reorientation toward field alignment. Here, the formation of chiral chains persists across a wide range of field strengths. The box plots illustrate that similar median orientations can arise from different combinations of s and α – for example, the median angle in s = 0.65 and α = 30 is identical to that in s = 0.75 and α = 70 (both ∼60°). This finding suggests that in this regime the magnetic field can be tuned to engineer specific structural configurations without necessitating precise control over dipole displacement. Worm-like and double-chain morphologies are characteristic of this region in the absence of an external field. In the case of low-field regime (α ≲ 10), these morphologies remain largely unchanged. For an intermediate-field regime (10 ≲ α ≲ 90), chiral chains dominate and maintain a central magnetic core. Finally, at high-field regime (α ≳ 90), resistance is completely diminished, and the structures collapse into field-aligned chains (Fig. 4n and o). Here, the high leverage of the shifted dipole magnifies torque imbalances, causing local competition between alignment and internal stability. The metastable structures are stabilized by dipole–dipole interactions that favor antiparallel or twisted configurations. As a result, the system becomes highly sensitive to small variations in α, with structure–function behavior governed by a fine balance between field-driven torque and geometrically induced frustration.

The median of the orientation distribution function as a function of α provides a clear metric to analyze how structures respond to increasing magnetic field strength (Fig. 5). For the low-shift regime, the median orientation remains constant at 1.0 across all values of α, indicating a persistent head-to-tail alignment, regardless of the presence or absence of a magnetic field. In this regime, particles readily form chains and loops that can be easily reoriented. The constancy of the median orientation in this regime reflects the low energy barrier for dipole realignment, which allows rapid structural reorganization even under weak external fields. These systems operate in a regime where thermal fluctuations are sufficient to overcome small misalignment penalties, enabling the system to remain near a global energy minimum.

Fig. 5 Evolution of the median orientation (m_i·m_j)_med/‖m‖² as a function of field strength α for various values of s. Low-shift systems remain aligned (median = 1.0) regardless of field. Intermediate and high-shift systems exhibit progressive reorientation. Notably, antiparallel double-chain structures at s = 0.75 show a transient drop in median before converging to full alignment at α ≳ 90.

A notable transition occurs for s = 0.5, where the median starts at 0.5 in the absence of a magnetic field and reaches 1.0 under magnetic field, indicating minimal resistance to reorganization. In contrast, at s = 0.6, which also belongs to the intermediate-shift regime, the system exhibits greater resistance to alignment. The median begins at −0.5 and only begins to increase beyond α > 20, eventually reaching 1.0 at high values of α. The contrast in reorganization kinetics between s = 0.5 and s = 0.6 can be attributed to a critical increase in asymmetry and the appearance of angular locking states that temporarily trap dipoles in suboptimal configurations. These barriers introduce hysteresis in the reconfiguration trajectory, visible as a delayed response in the median orientation.

The high-shift regime exhibits a more complex and progressive reorganization. In particular for s = 0.75, non-monotonic behavior is observed: at α = 20, the median orientation decreases to −0.6, deviating from the expected alignment trend. The observed non-monotonic behavior in the median orientation at s = 0.75 presents a notable departure from the otherwise progressive alignment trends observed across lower values of the dipolar shift. This anomaly is attributed to the formation of antiparallel double-chain structures: clusters in which two linear chains are arranged side by side with opposing dipole orientations, demonstrating significant resistance to field-induced reorientation. These clusters manifest parallel (+1.0) and antiparallel (−1.0) orientations; however, the magnetic field preferentially reorients only the parallel components. This type of antiparallel locking is analogous to domain wall pinning in magnetic materials,⁴¹ where local arrangements resist global magnetization until sufficient external energy is provided. The presence of aligned and anti-aligned domains in the same cluster produces a bimodal orientation profile, challenging the system's capacity for coherent response.

The proximity of the dipole to the surface of the particles at s ≳ 0.75 amplifies the torque experienced by the particles, rendering them highly sensitive to local field gradients while simultaneously making them susceptible to frustration due to steric hindrance and competing interaction minima.⁴² Energetically, these antiparallel configurations represent metastable states characterized by local minima in the magnetic potential landscape. The strongly anisotropic potential well at high s regimes leads to the localized trapping of particles in configurations that are not aligned with the global field direction but minimize dipole–dipole interaction energy. This phenomenon reflects a balance between competing energetic effects: alignment with the external field lowers the magnetic potential energy, while antiparallel dipole arrangements reduce pairwise dipolar interactions. The result is a frustrated system where alignment is delayed until the applied magnetic torque exceeds the stabilization provided by internal dipolar coherence.

As α increases further, the magnetic torque becomes sufficient to overcome this metastable stabilization, and the system transitions towards global alignment −0.5 at α = 30 and progressively increases until reaching a value of 1.0 for α ≥ 90. This delayed transition further underscores the complex interplay between particle-level anisotropy, field-induced torque, and collective rearrangement dynamics in the high-shift regime. It also suggests the presence of multiple competing energetic basins in the system, with potential relevance for tuning metastability in programmable self-assembled materials.

In particular, at α ≥ 90, all systems, regardless of s, converge to a median of 1.0, indicating universal alignment and a reduction in structural diversity.

3.4 Microstructural classification under magnetic field influence

The microstructure of MJPs is governed by the interplay between dipolar displacement s and the strength of the external magnetic field α, leading to a wide variety of self-assembled morphologies. The emergence of specific microstructures results from a torque balance between field-alignment forces and dipole–dipole interactions. This balance defines a potential energy landscape with multiple local minima, where the particle configuration depends on both the geometric asymmetry induced by s and the externally imposed field α. Thus, the microstructure reflects the equilibrium point—or metastable compromise—between competing alignment pathways.

To define the boundaries between the low-, intermediate-, and high-shift regimes, we follow the criteria established by Victoria-Camacho et al. (2020):⁹ the low-shift regime (s ≲ 0.35) is defined by head-to-tail aggregation with predominant simple chains and loops, and a decreasing aggregation rate (z) as s increases; the intermediate-shift regime (0.35 ≲ s ≲ 0.65) corresponds to bidirectional aggregation with island-like structures, where aggregation accelerates with increasing s; and the high-shift regime (s ≳ 0.65) features antiparallel double-chain aggregation with worm-like morphologies and comparatively slower aggregation kinetics.

These regime boundaries correspond to critical thresholds, where the system's response undergoes qualitative transitions. For instance, crossing from the low- to intermediate-shift regime corresponds to a bifurcation in the dominant aggregation pathway, from linear chain extension to multidirectional island formation. This shift is driven by an increasing angular offset between dipoles that disrupts the head-to-tail torque equilibrium.

Our results show that for low s, the system remains in low-energy head-to-tail configuration, exhibiting minimal resistance to reorganization. In contrast, for s ≳ 0.5, the high angular deviation and the increase in local dipolar moment generate frustration and topological constraints that stabilize the chiral or antiparallel double-chain states. The stabilization of these frustrated configurations arises from local energy wells created by spatial confinement and asymmetric torque fields. This effect is exacerbated as s increases, effectively locking particles into energetically suboptimal but geometrically favorable morphologies, such as chiral loops or double chains. These metastable configurations can only be reorganized when the magnetic torque exceeds a critical threshold.

A detailed analysis of orientation medians and aggregation rates (Fig. 6) reveals how microstructural evolution depends on s and α. In the low-shift regime, the clusters quickly align with the field direction across all values of α, leading to a stable median orientation of 1.0. However, the aggregation rate z varies subtly: for s = 0.0 and s = 0.1, z decreases slightly with increasing α due to the directional confinement of growth. In contrast, for s = 0.2 and s = 0.3, z increases as the looped morphologies collapse into linear chains. This collapse into linear chains can be interpreted as a field-induced structural phase transition. As α increases, the system continuously traverses a morphological phase space from disordered or isotropic arrangements to an anisotropic uniaxial configuration.

Fig. 6 Microstructure classification across dipolar shifts and field strengths. (a) Median orientation values highlight differences in alignment resistance across regimes. (b) Aggregation rate exponent z shows structural transitions from loops and bidirectional growth to field-aligned chains. Field intensity acts as a unifying parameter: at α ≥ 90, all systems converge to a common microstructure regardless of s.

In the intermediate-shift regime, the structural response becomes increasingly complex. At s = 0.4 and s = 0.5, the clusters initially form compact loops or islands that realign rapidly upon exposure to the magnetic field. However, at s = 0.6, loop-like structures have collapsed into bidirectional islands that exhibit resistance to realignment under low-field regime (α ≲ 10). This resistance is gradually overcome as the field strength progresses into the intermediate-field regime (10 ≲ α ≲ 90), leading to the formation of chiral chains and ultimately resulting in fully aligned morphologies.

The chiral chain region is therefore identified within this intermediate-field regime for intermediate and high shift, where the aggregates exhibit symmetric geometries with respect to the y-axis (field direction), despite being globally aligned. This behavior arises from residual alignment resistance and is characterized by a modest increase in the aggregation rate z with s.

In the high-shift regime, microstructural resistance becomes predominant. At low-field regime, clusters maintain their original complex double-chain or worm-like configurations. Only above α ≳ 10 begins the gradual structural reconfiguration, with full alignment achieved at high-field regime (α ≳ 90), where all systems, irrespective of s, converge into single chains aligned with the external field. This convergence highlights a field-induced universality in the aggregation behavior, overriding the intrinsic anisotropy imposed by the shifted dipole. The tangled chains, in particular, are observed at the interface between the chiral and fully aligned regions. These morphologies lack the y-axis symmetry typical of the chiral regime, but still exhibit partial resistance to alignment, suggesting an ongoing structural transition. Tangled configurations are also found at the transition boundaries between the low-, intermediate- and high-shift regimes (as defined by Victoria-Camacho et al. (2020)⁹) and the fully aligned state, particularly under weak or intermediate fields. Thus, these structures are inherently transient and structurally heterogeneous, and these tangled morphologies could serve as reconfigurable states in magnetically responsive materials. Their location at the boundary of morphological regimes makes them sensitive to slight variations in external fields, suggesting potential utility in tunable switching or sensing applications where structural adaptability is required.

Finally, in high-field regime (α ≳ 90), all systems converge to identical microstructures – fully aligned single chains – indicating a universal collapse of structural diversity driven by field intensity. This fully aligned chain region is defined for high magnetic fields (α ≳ 90) in all s, and also for intermediate fields (10 ≲ α ≲ 90) when s ≲ 0.5. In this regime, the structures consist of straight, field-aligned chains with uniform aggregation dynamics, and the system reaches a universal growth exponent of z ≈ 0.473.

In general, the observed morphological transitions across the (s, α) space are governed by a non-linear interaction between geometric anisotropy and torque-driven alignment. As s increases, the field strength required to overcome internal resistance also increases, but this requirement is mitigated once a critical α is reached, where the alignment torque dominates regardless of internal complexity. This leads to a convergence of the structural outcome and the dynamic behavior.

Table 1 offers a systematic classification of these morphological transitions, mapping typical structures observed in different combinations of s and α. In low magnetic field (α ≲ 10), the clusters exhibit various morphologies: head-to-tail chains with occasional loops in the low-shift regime, bidirectional islands in the intermediate-shift regime, and wormlike or antiparallel double chains in the high-shift regime. The application of an intermediate magnetic field (10 ≲ α ≲ 90) initiates structural reorganization: loop suppression and chain alignment in low-shift regimes and intermediate-shift regime at s ≲ 0.5, tangled and chiral chains formation in intermediate-shift regime at s ≳ 0.5 and high-shift regime. Finally, at high magnetic field (α ≳ 90) all systems converge in fully aligned single chains.

Table 1 Morphological classification of magnetic Janus particle (MJP) clusters as a function of dipolar shift s and magnetic field strength α. Structures range from loops and double chains in the absence of field to fully aligned single chains at high field intensity

α regime	s regime	Typical structures observed
Low field (α ≲ 10)	Low s (s < 0.35)	Head-to-tail chains with occasional loops and short rings
	Intermediate s (0.35 ≤ s < 0.65)	Bidirectional growth with island-like clusters
	High s (s ≥ 0.65)	Worm-like and antiparallel double chains
Intermediate field (10 ≲ α ≲ 90)	Low s	Fully aligned chains.
	Intermediate s	Tangled aligned chains.
	High s	Chiral and tangled chains with central magnetic cores and partial reorientation
High field (α ≳ 90)	Low s	Fully aligned chains.
	Intermediate s	Fully aligned chains.
	High s	Fully aligned chains.

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3.5 Field-induced structure diagram and predictive framework for high dipolar shift cases

The interaction between the field and particles, as previously documented, facilitates the realignment of existing structures, thus encouraging the development of single-chain, tangled-chain, and chiral configurations. Computational limitations render it impractical to numerically investigate systems characterized by substantial dipolar shifts s ≳ 0.75.

To elucidate the morphological transitions, we introduce the dimensionless parameter magnetic torque ratio (R_Mag), defined as R_Mag = T^H/T^P, which compares the deterministic Zeeman alignment torque (T^H) with the dipole–dipole reconstructive torque (T^P). Additionally, in the overdamped regime considered here, stochastic Brownian torques average to zero and primarily set kinetics (fluctuation-assisted barrier crossing), while viscous drag fixes the relaxation timescale; neither shifts the steady-state balance of deterministic torques. Consequently, the threshold condition for reconfiguration is governed by R_Mag alone with R_Mag only depend of T^H and T^P.

Upon non-dimensionalizing the relevant quantities, the dimensionless parameters α (field strength) and λ (dipolar coupling) naturally emerge (see 3. Derivation of the magnetic ratio R_Mag of the SI), leading to the following expression for R_Mag:

where represents the effective dimensionless distance between neighboring dipoles, accounting for the radial displacement from the particle centers and their relative orientation, as given by:

R _Mag indicates whether field alignment or internal reconstruction dominates the structural response: for R_Mag > 1 the field torque prevails and chains align with the field; for R_Mag < 1 the dipolar reconstructive torque stabilizes field-free morphologies. As derived previously, R_Mag increases with field strength α and effective interdipole spacing ,³ and decreases with the dipolar coupling λ and the change s. The denominator includes a lever-arm correction due to off-center dipoles, ensuring that R_Mag captures both force magnitude and torque arm in a single predictive metric.

In particular, in a typical head-to-tail configuration of simple chains, (m_i·m_j)/‖m‖² = 1, this results in . This indicates the limiting case in which the field torque predominates over realignment without substantial reconstruction. This critical threshold at R_Mag = 1 serves a role analogous to that of a phase boundary in classical thermodynamic systems. Below this point, internal interactions dominate and preserve intrinsic morphologies; above it, the system reorganizes into field-dominated states. This analogy offers a conceptual bridge to phase-transition theory and reinforces the idea that the structural response of MJPs is governed by order–disorder transitions mediated by torque asymmetry.

This methodology proves particularly beneficial when conventional simulation tools become unreliable because of numerical instabilities at high values s. The validity of the R_Mag criterion is corroborated by qualitative consistency with phase boundaries noted for laterally shifted dipoles,^10,12 and by recent predictive frameworks introduced by Tafur-Ushiñahua et al.⁴³ This ratio serves as a quantitative measure of whether field alignment or internal reconstruction dominates the system's structural response.

Fig. 7 presents the structure diagram constructed using the criterion R_Mag. The meshed region indicates the values computed using extrapolated parameters for (m_i·m_j)/‖m‖², as shown in Fig. S2 of the SI. The blue lines correspond to the condition R_Mag = 1 calculated by eqn (17), which appears in Fig. S3(a) of the SI. The orange line corresponds to R_Mag = 1 for the fixed value , shown in Fig. S3(b) of the SI.

Fig. 7 Structure diagram based on the R_Mag criterion. The x-axis represents the dipolar shift s, and the y-axis represents the Langevin parameter α. Discrete symbols indicate individual simulation outcomes for each regime single chain, loop like, island like, worm like, antiparallel double chain (ADC), aligned chains, tangled chain, and chiral chain. The orange curve corresponds to the assumption of a fixed inter-dipole distance , while the blue curve accounts for orientation-dependent distances. Six distinct morphological regimes are identified: in the black, blue, red, brown, and yellow regions, structures preserve their field-free morphology;⁹ in the green region, complete field-induced alignment into single chains occurs; in the purple region, tangled chains are observed; and in the olive-brown region, chiral chains exhibit partial resistance to alignment. The section with mesh indicates the extrapolated region; solid lines indicate values calculated by simulation, and dashed lines indicate extrapolated values.

The extrapolated region, particularly for s > 0.75, is indicated with a meshed overlay to emphasize that direct numerical simulations are not available due to unstable dynamics in these configurations. However, physical reasoning based on the R_Mag framework provides a consistent extrapolation that preserves observed scaling trends, allowing theoretical predictions in otherwise inaccessible parameter regimes.

The black, blue, red, brown, and yellow region at low values α corresponds to morphologies that are largely unaffected by the magnetic field, where the reconstructive torque predominates. In these regions, lower dipolar shifts require smaller magnetic field strengths to initiate realignment, while higher shifts necessitate stronger fields.

The green region signifies the complete dominance of field alignment, where all structures reorganize into chains oriented by a singular field.

The purple region shows a low dominance of field alignment, where all structures reorganize into chains oriented by a singular field. Structures formed near the blue boundary line experience competition between alignment and reconstructive torques, resulting in intermediate tangled chain morphologies that are neither fully aligned nor purely chiral. This region represent partially ordered structures where the system is energetically close to reconfiguration, but local minima in the potential energy landscape hinder immediate alignment. These intermediate tangled chains reflect a balance of torques that results in metastable arrangements.

The olive-brown region, emerging at s ≳ 0.5, hosts chiral chains. Here, the proximity of the dipole to the particle surface introduces significant resistance to field-induced alignment. Despite this, the trend indicates that sufficiently strong magnetic fields eventually overcome internal dipolar stabilization. Chiral chains emerge for intermediate/high s at intermediate α (olive-brown region in Fig. 7). Left/right degeneracy can be biased by: (i) seeded orientation (imposing a short aligned nucleus), (ii) a weak in-plane rotating component superimposed to H during ramp-up, and (iii) slight geometric or field asymmetries (gradients or boundary proximity). Increasing s broadens chiral stability, whereas increasing α raises the inversion barrier; thus s sets the window, α sets the lock.

The structure diagram in Fig. 7 constructed using R_Mag confirms that morphological transitions follow a universal progression driven by torque balance and provides a predictive criterion for high values of s where numerical instabilities preclude direct simulation. This framework allows for extrapolation beyond the current computational limits, grounded in physical quantities and validated trends. The separation into three morphological regimes—preserved structures, tangled chains, and field-aligned chains—offers critical insights into the tunability of cluster morphologies through external fields.

This structure diagram not only categorizes the morphologies, but also provides an operational framework for designing responsive materials. By selecting appropriate combinations of s and α, one can target specific structural outcomes, such as programmable anisotropy, reversible transitions, or metastable trapping. Consequently, R_Mag emerges as a design metric in the engineering of magnetically active colloidal systems.

Aligned single chains (green region) offer efficient translation under field steering; chiral chains (olive-brown region) enable controlled rotation/propulsion under AC or rotating fields; double chains provide stiffness and magnetic cores for loadbearing; islands/loops respond multiaxial but less directionally. For controllable motion of aligned chains, we recommend s ≲ 0.5 with moderate fields α ≃ 30–60 (steerable yet reconfigurable). For maximum rigidity/velocity choose α ≥ 90 (universal alignment).

4 Conclusions

This study investigates the dynamic aggregation behavior of magnetic Janus particles (MJPs) with radially shifted dipoles in quasi-two-dimensional systems. By systematically varying the dipolar displacement s and the external field strength α, we uncover six dominant microstructural regimes: (i) single-chain clusters and loop-like aggregates (low shift, low field), (ii) bidirectional islands (intermediate shift, low field), (iii) worm-like and antiparallel-double-chains (ADC) (high shift, low field), (iv) fully aligned field-directed chains (high field for all shift, or low shift with intermediate field), (v) tangled chains (close to R_Mag = 1), and (vi) chiral chains (intermediate to high shift, intermediate field). These morphologies arise from a competition between anisotropic dipolar interactions and magnetic field alignment, modulated by the shift-dependent torque asymmetry.

We redefine the role of field strength by introducing three distinct regimes based on the Langevin parameter: low-field regime (α ≲ 10), where structures largely preserve their intrinsic morphology; intermediate-field regime (10 ≲ α ≲ 90), where partially aligned or chiral states emerge; and high-field regime (α ≳ 90), where all systems universally converge into fully aligned chains. This convergence is quantitatively captured by the growth exponent z ≈ 0.473, marking a kinetic signature of complete structural alignment at all values of s.

Our analysis reveals that R_Mag serves as a robust predictor of structural transition, bridging microscopic orientation dynamics with emergent macroscopic order. Furthermore, the effective interaction distance provides a geometric link between local dipolar alignment and global morphology, allowing structural interpretation beyond qualitative observation.

Beyond identifying morphological transitions, these findings contribute to a deeper understanding of symmetry breaking and kinetic arrest in dipolar colloidal systems. The emergence of tangled and chiral morphologies in transition zones exemplifies the subtle interplay between field-induced constraints and internal anisotropy, highlighting new non-equilibrium pathways to complex structure formation. As such, this work advances the current framework of field-directed colloidal self-assembly and offers practical tools for designing adaptive, reconfigurable materials, with potential implications in soft robotics, targeted delivery, and programmable active matter systems.

Author contributions

J. A. V.-C.: conceptualization, methodology, software, investigation, visualization, writing – original draft. U. M. C.-F.: conceptualization, supervision, funding acquisition, writing – review and editing. All authors have read and approved the final manuscript.

Conflicts of interest

There are no conflicts to declare.

Data availability

Supplementary information including additional simulation details, numerical convergence tests, and supporting figures (finite-size analysis, extrapolated orientation statistics, and R_Mag threshold curves). See DOI: https://doi.org/10.1039/d5sm00872g.

All simulation data and analysis codes supporting the results of this study, including the datasets used to generate the figures and statistical analyses, are openly available at Zenodo under the DOI: https://doi.org/10.5281/zenodo.15483817.

Acknowledgements

This project was supported by the U.S. National Science Foundation under grants 1827894 and 2112550.

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