Understanding the Role of Particle Deformability on the Crystal and Glass formation using Two-dimensional Ring Polymer Model

TL;DR

The work addresses how deformability of soft particles controls crystal versus glass formation in two dimensions using a ring-polymer deformable-particle model with tunable angle stiffness $k_{\theta}$. It employs extensive molecular dynamics across density $\rho$ and temperature $T$ to analyze static and dynamic properties (e.g., MSD, $Q(t)$, $χ_4(t)$, van Hove functions) and structure (e.g., $\psi_6$, $g_6(r)$) as $k_{\theta}$ varies. Key findings show a deformability-induced amorphisation transition and, at higher density, a crystalline/hexatic regime emerging with larger $k_{\theta}$, alongside glassy dynamics characterized by $τ_α$, stretched exponent $β_{kww}$, MCT divergences at $T_{MCT}$, and VFT fits for $τ_α(T)$ and $D(T)$; Stokes–Einstein breakdown with $D \propto τ_α^{−κ}$ and $κ ≈ 0.74$, all accompanied by strong dynamic heterogeneity and growing static and dynamic length scales $ξ_s$ and $ξ_d$. Finite-size analyses reveal non-monotonic $τ_α(N_R)$ and concomitant length-scale growth, enabling data collapse and a $T$–$ρ$ phase diagram featuring amorphous and crystalline/hexatic regions. The deformable-ring framework provides a minimal, tunable platform to model confluent and non-confluent soft materials, with prospects for extension to 3D and core–shell architectures to capture complex rheology and yielding phenomena.

Abstract

Soft matter systems are common in nature and make up nearly all the essential components necessary for life, from cells to the organelles within those cells. The ability of these soft materials to deform is crucial for the proper functioning of various biological processes, such as blood flow in our veins and arteries. It is vital to understand how deformability influences the normal functioning of these processes. We have investigated an assembly of two-dimensional (2D) polymeric non-overlapping rings via extensive molecular dynamics simulations. The main idea is to study an assembly of model particles with anisotropic deformability using polymer rings. By tuning the degree of deformability of these model deformable particles, we study the dynamic and static properties of the assembly at different densities and temperatures. This deformable particle model might correspond to an assembly of epithelial cells or similar biologically soft bodies. In the limit at which the rings are very rigid with very little deformability, one expects to see the formation of a triangular lattice by the centres of these polymer rings. On the other hand, if one increases the deformability of these polymer rings, due to increased disorder, one observes glass-like dynamical behaviour even for identically sized polymer rings. We also show a transition from a crystalline state to a disordered glassy state driven solely by particle deformability. We observe non-trivial finite-size effects in the dynamics of these glass-forming ring polymers, not seen in usual molecular glass-formers.

Figures (8)

• Figure 1: Glassy Dynamics of Ring-polymer Assembly.(a) A typical configuration of the rings in a dense deformed state. One sees that the centre of mass (CoM) of these rings forms a disordered structure. (b) The radial distribution of the CoMs for two different system sizes ($N_R =289$ and $N_R=729$), which are practically indistinguishable. (c) The MSD plots for various temperatures for $k_{\theta}=10.0$ are plotted. The appearance of a clear plateau at lower temperatures indicates the onset of landscape-dominated glassy dynamics and caging effects. (d) The self-overlap functions, $Q(t)$, with stretched exponential fits show the hallmark two-step relaxation process typical of glass-forming liquids. The inset shows the temperature dependence of $\beta_{kww}$, the stretching exponent. (e) The VFT fits of $\alpha$-relaxation times at three different $k_{\theta}s$. (f) The corresponding diffusivity fits for the same $k_{\theta}s$ at a monomer density of $\rho = 0.23$.
• Figure 2: Correspondence with Mode Coupling Theory Predictions. (a) The fits for power law divergence of the relaxation times as predicted in MCT, (b) The same power law fits to $\tau_\alpha$ vs $T-T_{MCT}$. It is interesting to see that $\tau_\alpha$ obeys MCT predictions very well in the entire temperature window for all three values of $k_\theta$. (c) The fits to diffusivity ($D$) vs temperature follow the predicted MCT form. The fits are found to be very good. (d) The power-law fit to the $\chi_4(t)$ peaks vs. $\tau_\alpha$, as predicted by MCT ($N_R=289$). It becomes poor at lower temperatures, indicating a similar deviation seen in many molecular glass-forming liquids.
• Figure 3: Dynamical Heterogeneity. (a) A typical configuration of rings where the rings are coloured according to the distance moved in a time interval to demonstrate dynamic heterogeneity. (b) The increase in the peak height of the four-point dynamic susceptibility ($\chi_4(t)$) with decrease in temperature, (c) The van-hove function for different temperatures (Fits in the main plot are exponential fits and the plot in the inset are the Gaussian fits to the van-hove function), (d) The displacement field of the COMs for a specific time period which clearly shows the displacement fields are quite heterogeneous ($N_R=1089$). (e) The non-Gaussianity parameter ($\alpha_2(t)$) also shows the same trend as that of $\chi_4(t)$. (f) The breakdown of the Stokes-Einstein relation for three different $k_{\theta}$s. The $\kappa$ exponent in , $D\propto\tau_{\alpha}^{-\kappa}$, is $\approx 0.74$ with very slight change among $k_{\theta}$s (The data is obtained for $N_R=289$).
• Figure 4: Non-trivial Finite Size Effects. (a) The $\alpha$-relaxation times ($\tau_\alpha$) variation with system size for different temperatures. Notice the non-monotonic behaviour at an intermediate system size at lower temperatures (see text for detailed discussion). (b) The collapse of the relaxation times data for all the system sizes using a growing static length scale, $\xi_s$, as elaborated in the text, (c) The system size dependence of the $\chi_{4P}$s for different temperatures, which shows a clear sign of increase with system size. One observes some noise at intermediate system sizes, very similar to the system sizes where one sees non-monotonic behaviour of $\tau_\alpha$. (d) The $\chi_4$ peaks obtained by the block analysis method are discussed in the main text. Notice the improvement to the signal-to-noise ratio in this method. (e) The collapse of the scaled $\chi_4(L_B)$ with $L_B/\xi_d$. The data collapse is quite good, suggesting that the obtained length scale will have lower uncertainty. (f) The displacement-displacement correlation function, $\Gamma_{uu}(r,t_{\chi_{4P}})$, is plotted as a function of $r$ for all the studied temperatures. One clearly sees that the correlation increases with decreasing temperature, as the decay of the correlation function becomes slower. (f) The finite size effect of $\Gamma_{uu}(r,t_{\chi_{4P}})$ at $T=0.95$. There are no finite-size effects in $\Gamma_{uu}(r,t_{\chi_{4P}})$. (h) The growth of the dynamic and static length scales obtained from the different methods is plotted as a function of temperature. One clearly sees a strong growth of these length scales with decreasing temperature. (i) Cross plot of $\xi_d$ and $\xi_s$ to highlight the concomitant growth of these length scales in accordance with MCT-like dynamical behaviour with breakdown at lower temperatures.
• Figure 5: Reentrant Dynamical Crossover with System Size. (a) Shows the MSD for various system sizes, indicating a crossover in the dynamics from slower to faster to slower. (b) highlights the same crossover in the self-overlap correlation function. (c) Similar results for $\chi_4(t)$ with system size. All these analyses are done at $T=0.85$. At large system sizes, the $\chi_4(t)$ develops multiple peaks at short timescales and becomes noisy because of phonons. (d), (e) show the ASP and $R_g$ distribution for large and small system size, which show no difference, (f) The van Hove functions obtained at $t_{\chi_{4P}}$, the peak position of $\chi_4(t)$.