Rotational Jamming Transition

• Rotational jamming transition is defined as the kinetic arrest of angular motion in assemblies with orientational degrees of freedom, extending classical jamming concepts.
• The phenomenon arises in various systems—from dense packings of anisotropic objects to active spinners and optical matter—demonstrating unique zero-modes and scaling behaviors.
• Quantitative studies reveal that rotational jamming is marked by protocol-dependent contact numbers, distinct vibrational spectra, and critical scaling laws near the jamming threshold.

Rotational jamming transition refers to the kinetic arrest of angular degrees of freedom in assemblies with orientational, rotational, or internal shape variables. This phenomenon, which extends the classical idea of jamming in particulate matter, arises when constraints or crowding suppress not just translational but also rotational or internal dynamics. Rotational jamming manifests in diverse physical systems: particulate assemblies of anisotropic objects, active spinner or optical matter, frictional or frictionless suspensions under complex driving, and even in mean-field theories of glass transitions for systems with internal degrees of freedom. The transition is characterized by critical scaling laws, unique zero-modes associated with rotations, and, in many cases, distinct spectral signatures in the vibrational density of states.

1. Fundamental Definitions and Physical Mechanisms

In assemblies with non-spherical or internally-structured particles, the mechanical arrest at high density or strong interactions can propagate to rotational degrees of freedom as well as translational ones. Consider a system of particles with $d_f$ total degrees of freedom per particle (translations and rotations). Jamming, in this broader sense, occurs when the constraint network—generally encoded via interparticle contacts or enforced by external fields—renders motion impossible in one or more subspaces of the full configuration space.

Rotational jamming is thus identified by:

• Arrest of angular motion such that particle orientations become frozen ($\dot\theta_i\to0$ for all or a subset of particles);
• Emergence of associated zero-frequency modes (rotational rattlers or rotationally soft modes);
• Marked changes in rotational order parameters or angular velocities with increasing density or interaction strength.

This jamming can be driven by direct particle contacts (as in dense, aspherical packings), by interaction-driven torque balance (as in optically-driven systems), or by imposed protocols (e.g., alternating shear). The interplay between rotation and translation, and among various physical mechanisms, critically shapes the character of the transition and its universality class (Ikeda et al., 2019, Shiraishi et al., 2019, Shukla et al., 13 Nov 2025).

2. Rotational Jamming in Packings of Anisotropic and Rigid Bodies

In two or higher dimensions, particles with nontrivial geometry (e.g. dimers, stapled particles, ellipsoids) possess additional rotational degrees of freedom. Jamming transitions in such systems are distinguished from the isostatic scenario of spheres.

Contrary to frictionless spheres, which jam isostatically with a contact number $z_{\rm iso}=2d$, nonspherical objects in $d$ dimensions require

$z_{\rm iso} = 2(d + d_{\rm rot})$

with $d_{\rm rot}$ the number of independent rotational degrees of freedom per particle. However, extensive simulations and mean-field theory show that true packings of such objects are typically hypostatic at nominal jamming, i.e., $z_{J} < z_{\rm iso}$, unless careful correction is made for rotational rattlers—particles whose centers are translationally immobilized but which retain a free rotation (Shiraishi et al., 2019).

For dimer packings, explicit analysis reveals that after excising all rotational rattlers, the corrected contact number converges to $z_c=2d_f$ in the thermodynamic limit, restoring isostaticity. The excess contact number above jamming, $\Delta Z = Z-Z_c$, and the onset frequency of rotational soft modes, $\omega_{\rm rot}$, both vanish as $\sqrt{\Delta\phi}$ or $p^{1/2}$ near the transition:

$$\omega_{\rm rot} \propto \Delta Z \propto \sqrt{\Delta\phi} \propto p^{1/2}$$

where $\Delta\phi = \phi - \phi_J$ and $p$ is pressure (Shiraishi et al., 2019). The vibrational spectrum exhibits two plateaus—one associated with translational modes, another with rotational—separated by a "rotational peak" set by bare rotational stiffness.

The interplay between rotational and translational constraints thus generates a two-band marginal spectrum near jamming, a universal scenario for aspherical finite objects (Marschall et al., 2014, Shiraishi et al., 2019).

3. Rotational Jamming in Driven and Active Systems

The rotational jamming transition also manifests in assemblies subject to non-equilibrium driving or active forces. In systems of active spinners (self-driven dimers subjected to constant internal torque), the transition divides three dynamical regimes:

1. Absorbing state ($\phi < \phi_{c1}$): All dimers rotate freely, $\bar\Omega \to 1$ (normalized by the free-spin rate), with vanishing jammed fraction $f_j$.
2. Locally-jammed regime ($\phi_{c1}<\phi<\phi_{c2}$): A coexistence of non-spinning, locally jammed clusters and freely spinning dimers. Here, $f_j$ grows as $f_j \sim (\phi-\phi_{c1})^{1/2}$, and the standard deviation of spin rates peaks, marking phase coexistence.
3. Fully jammed state ($\phi>\phi_{c2}$): All angular and translational motion is arrested, $\bar\Omega\to0$ (Liu et al., 2022).

This transition is fundamentally collective, with caging and cooperative blocking. The local rotational jamming is signaled by the emergence of non-spinning clusters whose size diverges at threshold, reminiscent of typical glassy or jamming criticality.

Notably, the transition also produces hyperuniform phases in the absorbing (Class III) and locally-jammed (Class I) regimes as characterized by the decay of density fluctuations and the structure factor in reciprocal space. These features link rotational jamming to broader themes in disordered matter and materials design.

4. Shear, Alternating Protocols, and Friction—Control of Rotational Jamming

In suspensions of spheres subjected to time-dependent driving, such as alternating shear with rotation of the flow direction by an angle $\theta$, the jamming transition can be controlled via the amount and nature of angular rearrangement (Acharya et al., 16 Mar 2025).

For frictionless particles ($\mu=0$), the jamming point packing fraction $\phi_j(\theta)$ is invariant to protocol: $\phi_j(\theta) \approx 0.64$ independent of $\theta$. For frictional particles ($\mu=1$), $\phi_j(\theta)$ increases monotonically with $\theta$, from $\approx 0.59$ at $\theta=0$ to $\approx 0.62$ at $\theta=180^\circ$. This is attributed to the breaking and re-orienting of anisotropic force chains, which must reform after each rotation, necessitating a higher $\phi$ for percolating contact networks.

Hydrodynamic (lubrication) and contact stresses both contribute to the total shear stress:

$$\eta_h(\phi,\theta)/\eta_f \approx \alpha(\theta) (\phi/\phi_m)^{\beta(\theta)}$$

$$\eta_c(\phi,\theta)/\eta_f = \eta_0(\theta)\left[1-\phi/\phi_j(\theta)\right]^{-2.5\phi_j(\theta)}$$

As $\theta$ increases, contact stresses are weakened, but hydrodynamic stresses rise, shifting the regime of dominant dissipation.

Mean contact number $Z$ tracks the approach to jamming, decreasing with increasing $\theta$, and the mapping between $Z$ and $\eta_c$ collapses onto universal master curves for frictionless and frictional systems. The protocol-dependent tuning of rotational rearrangements hence directly impacts the mechanical threshold and dissipation pathways (Acharya et al., 16 Mar 2025).

5. Order Parameters, Zero-Modes, and Universality Classes

Rotational jamming transitions are characterized by distinct order parameters and scaling laws:

• Orientational order: Quantified via nematic ($S_2$) and tetratic ($S_4$) parameters. In shear-driven systems of concave staples, $S_2$ (alignment with flow) drops to zero at the jamming point, while $S_4$ (90$^\circ$ order) rises just above it, reflecting local orientational motifs (Marschall et al., 2014).
• Angular velocity: The mean particle angular velocity $\bar\omega(\phi)$ under shear rises as $\phi$ increases, saturates just below jamming, and then increases further above $\phi_J$ due to anisotropic elastic torques.
• Contact number and isostaticity: In the mean-field model, the contact number at jamming for nonspherical particles is hypostatic, $z_J=1 + O(\sqrt{\Delta})$, with $\Delta$ encoding particle asphericity. The excess contact and modulus vanish linearly with $\delta\phi$ for $\Delta>0$, distinct from the sphere case, and associated distributions of gaps, forces, and vibrational densities change character (Ikeda et al., 2019).
• Zero-modes and "rotational rattlers": Packings may harbor localized zero-energy rotations ("rattlers") that do not contribute to mechanical rigidity. The correct counting of contacts must remove these, preserving the isostatic principle in the effective constrained subspace (Shiraishi et al., 2019).

The universality class of rotational jamming, especially in mean-field theory, is distinguished from the classical hard-sphere case by hypostaticity and linear scaling laws, arising directly from the presence and role of rotational or internal variables (Ikeda et al., 2019, Yoshino, 2017).

6. Rotational Jamming in Optical Matter and Emergent Symmetry-Breaking

Assemblies of plasmonic nanoparticles in circularly polarized light ("optical matter") exemplify a driven, nonthermal setting for rotational jamming (Shukla et al., 13 Nov 2025). Here, rotational arrest is not due to mechanical crowding but to the loss of discrete rotational symmetry necessary to harness spin angular momentum (SAM) from the light field.

Key features include:

• Torque transfer efficiency: The net optical torque

$$\tau = N f_{\rm SAM}(I, w_0, R) \approx C I \exp(-R^2/w_0^2)$$

rises with particle number $N$ only up to a critical cluster size set by the beam waist $w_0$.

• Symmetry order parameter: The bond-orientational order parameter $\psi_6$ (for hexagonal order) is nearly unity in stable rotors and plummets upon adding a "defect" particle, correlating with stagnation of rotation.
• Critical behavior: The transition occurs as cluster radial extent $R$ approaches $w_0$, with rotation ceasing sharply due to symmetry disruption.

This symmetry-driven kinetic arrest is directly analogous, in the rotational domain, to translational jamming, but is governed by the commensurability between assembly geometry and the driving optical field. The platform enables exploration of jamming, clogging, and kinetic arrest phenomena unique to rotationally active, driven nanostructures (Shukla et al., 13 Nov 2025).

7. Broader Theoretical Frameworks and Mean-Field Insights

Extensions to quantum, glassy, and abstract vector-spin settings further clarify the scope of rotational jamming. In exactly solvable mean-field models of M-component vector spins with multi-body interactions, glass transitions and jamming transitions emerge even without quenched disorder (Yoshino, 2017). Introducing hardcore (exclusion) potentials in "spin space" produces SAT/UNSAT (jamming) transitions that are formally isostatic, with power-law criticality matching hard-sphere models. These systems exhibit replica symmetry breaking and self-generated randomness.

The mean-field theory for jamming with internal/rotational degrees (e.g., the polydisperse-perceptron model) produces hypostaticity, linear scaling of excess contact and modulus, and a new universality class for jamming transitions distinct from the $\delta z \sim \delta\phi^{1/2}$, $G\sim \delta\phi^{1/2}$ regime of spheres (Ikeda et al., 2019).

The interplay of rotational and translational jamming, the role of zero-modes, and the dependence upon protocol and symmetry, thus define a rich taxonomy of arrested states in classical and driven assemblies. The rotational jamming transition provides a concrete paradigm for exploring these effects quantitatively across dimensions, geometries, and dynamical regimes.