Structure and dynamics of a Rouse polymer in a fluctuating correlated medium

TL;DR

The authors model a harmonic-confined Rouse polymer coupled to a fluctuating Gaussian field to capture memory effects from a correlated medium. They derive an exact effective description in the weak-coupling limit, revealing a non-Markovian generalized Langevin equation for the center of mass and linearized dynamics for the higher Rouse modes, with a memory kernel that decays algebraically when slow field modes are present. Key findings show algebraic relaxation of the center of mass near criticality or under conserved dynamics, accelerated relaxation of internal modes, and a medium-driven reduction of the polymer size as field correlations grow. The work also quantifies force-extension behavior and nonequilibrium size under driving, highlighting how field-mediated attractions modulate polymer mechanics and suggesting broader implications for polymers in complex, fluctuating environments.

Abstract

We study the static and dynamical properties of a harmonically confined Rouse polymer coupled to a fluctuating correlated medium, which affect each other reciprocally during their stochastic evolution. The medium is modeled by a scalar Gaussian field which can feature modes with slow relaxation and long-range spatial correlations. We show that these modes affect the long-time behavior of the average position of the center of mass of the polymer, which, after a displacement, turns out to relax algebraically towards its equilibrium value. This is a manifestation of the non-Markovian nature of the effective evolution of the position of the center of mass, once the degrees of freedom of the medium have been integrated out. In contrast, we show that the coupling to the medium speeds up the relaxation of higher Rouse modes. We further characterize the typical size of the polymer as a function of its polymerization degree and of the correlation length of the medium, particularly when the system is driven out of equilibrium via the application of a constant external driving force. Finally, we study the response of a linear polymer to a tensile force acting on its terminal monomers.

Figures (9)

• Figure 1: Schematic illustration of the model, consisting of a (linear) polymer chain coupled to a thermally fluctuating order parameter field $\phi(\bm{x},t)$, and confined by a harmonic potential with stiffness $\kappa_c$.
• Figure 2: Equilibrium distribution of the distance $|X_2-X_1|$ between the two sub-units of a dimeric molecule at positions $X_{1,2}$, in $d=1$ and in the absence of confinement. Solid lines represent the theoretical prediction in Eq. \ref{['eq:marginal_distribution_polymer']} specialized to the case $N=2$, whereas symbols are obtained with molecular dynamics simulations (see App. \ref{['sec:simulations']} for details). Left panel: the monomers have the same interaction coupling with the fluctuating field, i.e., $\sigma_1=\sigma_2$. This induces a collapse of the dimer, which is increasingly more effective as the field approaches criticality (i.e., as $r\to 0$). Right panel: the monomers have opposite interaction couplings, i.e., $\sigma_1= - \sigma_2$, resulting in a repulsive interaction and therefore in a stretching of the dimer. In the simulation we chose $\lambda=10$, while all other parameters were set to unity.
• Figure 3: Relaxation time $\hat{\tau}_{1,2}$ of the Rouse modes $\bm{\chi}_1$ and $\bm{\chi}_2$ as functions of the deviation $r$ from criticality. The plot shows that the coupling with a fluctuating and correlated medium speeds up the relaxation of the internal structure of the polymer. This effect is more pronounced when the field is characterized by correlations on large length scales (i.e., as $r \to 0$). The theoretical prediction (solid line) $\hat{\tau}_j \equiv 1/[\tilde{\gamma}_j+\Gamma(0)]$ has been obtained on the basis of Eqs. \ref{['eq:eff_lin_dyn_ho_modes']}, \ref{['eq:relaxation_rates_polymer']}, and \ref{['eq:lin_mem_kernel_polymer']}. The simulation results (symbols), instead, were obtained by extracting the slopes $1/\hat{\tau}_{1,2}$ of the relaxation of the Rouse modes (obtained via molecular dynamics simulations) in logarithmic scale via a linear fit (see Fig. \ref{['fig:linear_fitting']}). The two lines tend to their respective asymptote $1/\tilde{\gamma}_{1,2}$. This plot was obtained with a linear polymer of $N=10$ monomers, while all other parameters were set to unity.
• Figure 4: Nonequilibrium relaxation of the average center of mass of a linear polymer initially displaced from the rest position of the trap by an amount $\bar{\chi}_0$, in spatial dimension $d=1$. The symbols indicate the result of numerical simulations (see App. \ref{['sec:simulations']} for details), with the field initialized in the flat configuration $\phi_{\bm{q}}=0$ for all momenta $\bm{q}$. As specified in the legend, each color corresponds to a different correlation lengths $\xi_\phi=r^{-1/2}$ of the field. Left and right panels report the relaxation of $\langle \chi_0 \rangle$ obtained with the field evolving according to model A or model B dynamics, respectively. The algebraic decay of this quantity at long times, theoretically predicted in \ref{['eq:relax_modelA', 'eq:relax_modelB']}, is indicated by the dashed lines. The plots show that the relaxation of the center of mass towards the bottom of the confining potential is slowed down by the order parameter field, especially when the latter is critical. In the simulation we used $N=10$ and $T=0.01$, while all other parameters were set to unity.
• Figure 5: Typical size of a linear polymer at equilibrium, quantified by the mean-square gyration radius $\langle R_g^2\rangle$ (see Eq. \ref{['eq:def_Rg']}) in the absence of stretching forces. The results of the numerical simulations in $d=1$ (grey symbols) are compared to the theoretical predictions obtained with either the weak-coupling approximation (red lines, see \ref{['eq:Rg_correction']}), or the linearized theory (light blue lines, see \ref{['eq:Rg_linearized_theory']}). The choice of parameters in the main plots is such that the weak-coupling approximation is more accurate than the linearized theory. The two insets show instead that, with higher coupling $\lambda$ and lower temperature $T$, the linearized theory is more accurate. Left panel: $\langle R_g^2\rangle$ of a linear chain with polymerization degree $N=20$ as a function of $r=1/\xi^2_{\phi}$. The value of $\langle R_g^2\rangle$ is measured in units of $\bar{R}_g^2$, i.e., of the gyration radius of a free chain (i.e., in the absence of interaction with the medium) with the same parameters (green dashed line). This value is given in \ref{['eq:unperturbed_Rg2']} upon setting $\bm{f}_s=\bm 0$. The figure shows that, when the field approaches the critical point $r=0$, the typical polymer size is reduced as a consequence of the larger field-mediated forces. In the main plot we used $\lambda=0.3$ and $T=1$, while in the inset we chose $\lambda=1$ and $T=0.1$ (all other parameters were set to unity in both cases). Right panel: $\langle R_g^2\rangle$ of a linear chain as a function of $N$, measured in units of the bond length $l_0=dT/(\kappa + \kappa_c)$. For a sufficiently large $N$, the field-mediated forces induce a collapse of the chain, as shown by the non-monotonic behavior of the curves. The simulation parameters are the same as in the left panel, with $r=1$.