Active Polymers: Excluded Volume Effects

• Active polymers with excluded volume are chain-like systems that combine internal driving forces with non-crossing interactions to yield novel static and dynamic behaviors.
• They display nonmonotonic conformational responses where intermediate activity compresses the chain and high activity promotes swelling through modified scaling laws.
• Simulation studies reveal regime-dependent dynamics, including accelerated relaxation, altered diffusion, and robust rotational motion in ring architectures.

Active polymers with excluded volume comprise systems in which chain-like molecules are endowed with internal or external driving forces, and individual monomers interact via non-crossing (self-avoiding) repulsions. Unlike passive polymer models, active polymers introduce persistent mechanical agitation at the monomer level—commonly modeled as self-propulsion or tangential forcing—which, together with excluded-volume (EV) effects, generates qualitatively novel static and dynamic behavior. Recent simulation and theoretical studies elucidate a rich regime structure, with nonmonotonic conformational responses, regime-dependent dynamical scaling, and robust long-time rotational phenomena in ring architectures. Notably, excluded-volume dramatically alters both conformational scaling and dynamical crossovers induced by activity, preventing collapse and enabling swelling at sufficiently high activity.

1. Model Formulation and Core Control Parameters

Active polymers are modeled as bead-spring chains—either linear or closed-ring—where each monomer experiences stochastic thermal kicks, conservative bonding and EV forces, and a persistent active component. Excluded-volume is captured via truncated Lennard-Jones (Weeks–Chandler–Andersen, WCA) potentials for nonbonded pairs, ensuring short-range repulsion and preventing chain self-crossing. The primary sources detail Hamiltonians for bond stretching ($U_\text{bond}$), bending rigidity ($U_\text{bend}$), and EV interaction ($U_\text{ex}$), as

$$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$

with $\Theta$ the Heaviside function. The typical dimensionless activity measure is the Péclet number, $Pe = F_a l_0 / (k_B T)$ (or analogously for tangential drive), reflecting the active force strength relative to thermal agitation.

Key dimensionless parameters include:

• $Pe$: Activity strength
• $\ell_p / L$: Ratio of persistence length to contour length (stiffness, for rings)
• $\varepsilon / k_B T$: EV interaction strength
• $Wi$, $U_\text{bend}$0: Weissenberg number, for active chains in shear
• Chain topology: linear, closed ring

Both underdamped (Newtonian + Brownian multiparticle collision thermostat) and overdamped (Langevin) simulation regimes appear in the literature, depending on system and observable of interest (Lamura, 2024, Panda et al., 2023, Anand et al., 2020).

2. Conformational Scaling and Regime Crossovers

The effect of excluded volume and activity on conformational statistics, specifically end-to-end ($U_\text{bend}$1) and gyration radii ($U_\text{bend}$2), features strong nonmonotonicity as $U_\text{bend}$3 is varied. Three distinct regimes are established for flexible chains (Anand et al., 2020):

• Passive regime ($U_\text{bend}$4): $U_\text{bend}$5 with $U_\text{bend}$6 (Flory exponent for 3D self-avoiding walks).
• Intermediate activity ($U_\text{bend}$7): The chain exhibits compression, i.e., $U_\text{bend}$8 decreases relative to passive. This shrinkage arises from activity-induced enhancement of local monomer packing, evidenced by peaks in radial distribution functions and increased coordination numbers.
• Strong activity ($U_\text{bend}$9): EV steric constraints result in swelling, $U_\text{ex}$0 (active Rouse scaling), with Flory exponent $U_\text{ex}$1. In this regime, the chain conformation approaches that of a non-ev chain with strong activity.

The critical roles of EV are underscored in topologically distinct geometries:

• Active ring polymers: In the absence of EV (“phantom” rings), increasing $U_\text{ex}$2 yields pronounced shrinkage ($U_\text{ex}$3 drops by $U_\text{ex}$4 at $U_\text{ex}$5), while for self-avoiding rings, $U_\text{ex}$6 grows moderately ($U_\text{ex}$7 increase). For semi-flexible and stiff rings ($U_\text{ex}$8), conformational size becomes essentially independent of $U_\text{ex}$9 as rigidity dominates (Lamura, 2024).

For polymers subjected to external flow, such as shear, the EV–activity interplay yields a nonmonotonic extension: at moderate activity, active "kicks" perpendicular to flow promote compression; at high $$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$0 the chain re-expands and the scaling exponents for extension and alignment change, with $$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$1 and alignment $$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$2, distinct from passive values (Panda et al., 2023).

3. Structural Correlations and Local Packing

Structural observables—radial distribution functions $$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$3, coordination numbers $$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$4, collision times $$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$5—reveal the microscopic underpinnings of the chain's non-monotonic conformational response to activity. Simulations show:

• In the compression regime ($$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$6), $$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$7 exhibits enhanced first and second peaks, reflecting higher local crowding. The coordination number increases, and average collision time $$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$8 between monomers drops, as proximity events become frequent.
• For $$U_\text{ex}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 + \frac{1}{4} \right]\Theta(2^{1/6}\sigma - r),$$9, $\Theta$0 peaks diminish below the passive baseline, $\Theta$1 falls, and $\Theta$2 rises, consistent with chain stretching and reduced crowding (Anand et al., 2020).
• Energetically, per-monomer excluded-volume contributions peak at intermediate $\Theta$3, then decline as the chain swells; bond stretching increases with activity.

For ring geometries, distribution functions $\Theta$4 narrow with increasing $\Theta$5, with their modes shifting to smaller (phantom) or larger (self-avoiding) values. In a narrow stiffness window, phantom rings access multipeak states (e.g., transient double-ring conformations) (Lamura, 2024).

4. Dynamical Properties: Relaxation and Transport Modes

Activity and excluded-volume produce distinct dynamical signatures:

• Relaxation time $\Theta$6: The longest-mode relaxation, from end-to-end vector autocorrelation, decreases with activity in a two-stage power law:
  • Intermediate regime ($\Theta$7): $\Theta$8 (faster than passive Rouse).
  • High activity ($\Theta$9): $Pe = F_a l_0 / (k_B T)$0 (matching active Rouse scaling) (Anand et al., 2020).
• Centre-of-mass diffusion: $Pe = F_a l_0 / (k_B T)$1, with $Pe = F_a l_0 / (k_B T)$2.
• Segmental mean-square displacement (MSD): Shows subdiffusive scaling $Pe = F_a l_0 / (k_B T)$3 with $Pe = F_a l_0 / (k_B T)$4 at intermediate times, with this window narrowing as $Pe = F_a l_0 / (k_B T)$5 increases.
• Ring polymers:
  • Phantom active rings: Display activity-enhanced diffusive MSD at large $Pe = F_a l_0 / (k_B T)$6.
  • Self-avoiding active rings: Exhibit ballistic MSD ($Pe = F_a l_0 / (k_B T)$7) at intermediate times, with oscillatory modulations set by the rotational period (Lamura, 2024).
  • Internal dynamics is thus richer with EV, supporting long-range ballistic modes absent in phantom models.

In presence of shear, EV and activity result in nontrivial rheology and orientation dynamics. The zero-shear viscosity $Pe = F_a l_0 / (k_B T)$8 for EV-polymers is nonmonotonic with $Pe = F_a l_0 / (k_B T)$9—initially decreasing due to activity-induced compression, later increasing ($Pe$0) as swelling dominates (Panda et al., 2023).

5. Long-Time and Collective Modes: Tank-Treading and Rotation

Above a threshold $Pe$1, active rings develop persistent “tank-treading” rotation—steady circulation of monomers around the ring backbone. The rotation period $Pe$2 scales as $Pe$3, or more precisely $Pe$4; this scaling is robust with respect to both stiffness and excluded-volume strength (Lamura, 2024).

The rotational dynamics, quantified via ring diameter autocorrelation or direct measurement of bead trajectories, demonstrate that at long times the angular velocity becomes independent of bending rigidity or EV. Thus, excluded-volume, while impactful on internal structure and swelling, does not affect the global rotational mode in strongly-driven rings.

6. Physical Insights and Regime Diagrams

The interplay of active dynamics and excluded volume leads to the following physical picture:

• Excluded volume prevents activity-induced collapse seen in phantom models, yielding chain swelling—rather than shrinkage—at high $Pe$5 (Lamura, 2024, Anand et al., 2020).
• The non-monotonicity of chain size as a function of $Pe$6 is a direct consequence of EV: intermediate activity compresses the chain, while stronger propulsion overcomes steric constraints, restoring swelling but with modified scaling exponents.
• Dynamical crossovers in relaxation time ($Pe$7, switching from $Pe$8 to $Pe$9) and segmental MSD reflect the evolving balance of entropic elasticity, activity, and steric repulsion.
• Under flow, activity enables tuning of macroscopic rheology: moderate $\ell_p / L$0 softens flow-stretching and reduces viscosity, while strong $\ell_p / L$1 leads to isotropic swelling and Newtonian behavior at high strain rates (Panda et al., 2023).

Summary regime diagrams:

$\ell_p / L$2 Range	Flory Exponent $\ell_p / L$3	$\ell_p / L$4 Trend	Relaxation Time $\ell_p / L$5
$\ell_p / L$6	$\ell_p / L$7 0.588	$\ell_p / L$8	$\ell_p / L$9 (slow)
$\varepsilon / k_B T$0	minimal ($\varepsilon / k_B T$1)	$\varepsilon / k_B T$2	$\varepsilon / k_B T$3 (fastest)
$\varepsilon / k_B T$4	crossover	$\varepsilon / k_B T$5	crossover to $\varepsilon / k_B T$6
$\varepsilon / k_B T$7	$\varepsilon / k_B T$8	$\varepsilon / k_B T$9	$Wi$0 (active Rouse scaling)

7. Significance, Applications, and Outlook

Active polymers with excluded volume serve as minimal but versatile models for cytoskeletal filaments, synthetic active macromolecules, and “smart” rheological modifiers. The nontrivial conformational and dynamical effects induced by activity–EV interplay suggest avenues for active control of material properties unavailable in passive systems.

Key findings—activity-driven conformational non-monotonicity, EV-controlled ballistic transport, and robust collective rotation—provide mechanistic insight into behaviors of biological and synthetic polymers in active environments. The parameter-dependent regime diagrams offer guideposts for tuning activity, chain length, stiffness, and topology to target specific dynamical responses or material functionalities (Lamura, 2024, Panda et al., 2023, Anand et al., 2020).