Conformation and dynamics of ring polymers under symmetric thin film confinement

The molecular architecture of polymers is a key factor that affects the conformational, dynamic, and viscoelastic properties of polymer melts. For linear polymers, the “tube-model” and Rouse model have been well established to describe the motion of polymers with and without entanglements, respectively. For branched polymers, due to the entropy barrier between different conformations, the longest relaxation time exhibits exponential growth with strand length, which is explained by the arm retraction model.^1–3 Ring polymers, unlike linear and branched polymers that have free chain ends, exhibit distinct physical properties in polymer melts due to their topology, which reduces the number of conformations available to the molecule. Molecular simulations have been a powerful tool in studying the static (structural and conformational), dynamic, rheological, and topological (e.g., threading events) properties of ring polymer melts.^4–15 

Recently, ring polymers have attracted attention not only in fundamental polymer science but also in biophysics because DNA is circular in many biological systems, such as bacterial and mitochondrial DNA in eukaryotic cells. When strongly confined within a cell nucleus, chromosomes show a high level of spatial organization with DNA molecules occupying distinct territories, which is analogous to a melt of non-concatenated ring polymers.^16–19 Jung et al. modeled the bacterial chromosome by simulating a ring polymer under cylindrical confinement and observed an interplay between chain topology and confinement to explain the long-standing observations of chromosome organization and segregation in E. coli.²⁰ Another study has also found an analogy between the properties of confined polymers and that of chromosomes in bacteria where the conformational entropy is reduced when the polymers are more compact, inducing the chain segregation.²¹ Therefore, understanding the conformation, segregation, and dynamics of ring polymers under strong confinement is important not only for polymer science but may also have relevance to the study of chromosomes and DNA organization.

Linear polymers under confinement, which have been widely studied both computationally and experimentally, provide a foundation for our study of behavior of ring polymers.^22–35 It has been widely observed that a significant reduction of the entanglement density of linear polymers induced by nanoscale confinement can accelerate the dynamics of entangled polymers under various confinement geometries, including nanoscopic pores, thin films, and nanocomposites.^36–43 In recent work,⁴⁴ we demonstrated competing effects between the local friction near the wall and chain disentanglement in linear polymers confined within diamond-like struts. Previous studies on ring polymers have focused primarily on the static properties under confinement;^20,45,46 however, studies examining changes in dynamic properties are limited, and strongly confined systems have been given less attention. Recently, Pressly et al. demonstrated a strong correlation hole effect induced by 2D (cylindrical) confinement that causes linear polymer chains to segregate from each other under strong confinement,³⁶ an effect that is much weaker in planar thin films.⁴³ The correlation hole effect under 2D confinement led to a non-monotonic dependence of the diffusivity on the extent of confinement; an initial increase in the diffusivity associated with disentanglement was followed by a sharp decrease due to the enhanced correlation hole effect. Since ring polymers exhibit a more pronounced correlation hole effect compared to linear polymers,¹⁰ we hypothesize that the confinement-induced correlation hole effect requires less confinement in ring polymers compared to linear polymers.

In this work, we focus on unknotted ring polymer melts under symmetric thin film confinement and show the effects of confinement on both chain conformations and dynamics over a range of chain lengths and film thicknesses. Compared to linear polymers, we observe a stronger center of mass layering of ring polymers in the direction normal to the film. The changes in the average chain conformations in both perpendicular and parallel directions to the wall are similar to that of the linear polymers. We observe a monotonic decrease in the diffusivities as the film thickness decreases due to the increased role of friction between the polymers and the walls, which is supported by a Rouse mode analysis of our ring polymers. Finally, we demonstrate that the effect on the segmental dynamics extends over several monomer sizes.

Our molecular simulations employ the coarse-grained bead–spring Kremer–Grest (KG) model,⁴⁷ where non-bonded monomers interact through the repulsive part of the Lennard-Jones (LJ) potential: $Ur=4εσr12−σr6$ for $r≤rcut=216σ$⁠. All quantities are reported in units that are normalized by the potential strength, ε, the monomer size, σ, and the time $τ=σ(m/ε)12$⁠, where m is the monomer mass. The bonded interactions connecting two successive monomers are governed by a finitely extensible nonlinear elastic (FENE) potential with k = 30ε/σ² and R₀ = 1.5σ. The polymer model we use in this study corresponds to the ideal situation of fully flexible polymers, and the number of monomers per chain in our simulations is N = 50, 100, 200, or 350. For linear polymers employing this flexible model, the average number of monomers between entanglements is N_e ≈ 50. All the simulations are performed with the Large-scale Atomic/Molecular Massively Parallel Simulator molecular dynamics (MD) simulation package with the velocity-Verlet algorithm and a Nosé–Hoover thermostat.⁴⁸ The temperature is constant at T = 1.0 in all simulations presented below.

To construct the symmetric thin films, we begin with a rectangular box with a z-dimension of the height of thin film thickness H + 4σ, and the box is filled with LJ particles at a density of ρ_s = 1.3/σ⁻³. The system is equilibrated in the NVT ensemble at T = 1.5 where the fluid is in a liquid state. After equilibration, we remove the particles in the center of the box to create a cavity of the desired film thickness with 2σ-thick walls on either side of the box of amorphous LJ particles. The interaction between the wall beads and the polymers is identical to that between the polymer monomers, and the wall monomers are not integrated during the production simulations. Thin films with five different thicknesses (H = 5, 7, 10, 14, 20σ) are created to cover a range of confinement ratios (Fig. 1).

To insert the ring melts into the cavity, we separately begin with a large empty simulation box, of which the sizes on x and y directions are the same as thin films constructed previously. Ring polymers are placed on a cubic lattice in this box at a low density to avoid any concatenation, and each molecule is constructed by placing monomers on a circular path in a plane with a random orientation. As such, the initial density of the system is very low since the rings are widely separated. We start with an NVT simulation to relax the initial configurations, and we subsequently compress the box in the z-direction at a slow rate to ensure the stability of the simulation during compression. The compression continues until the box can fit into the thin film cavity described above, and the bead density inside the cavity will be ρ_s = 0.85/σ⁻³, as is commonly used for coarse-grained simulations. During the compression, the boundary condition in the xy-directions is periodic and the z-direction is set as non-periodic. Finally, we insert the ring polymers between the amorphous walls generated as described above, after which we begin the equilibration process. We equilibrate at constant temperature using standard MD until the center of mass mean-squared displacement (MSD) exhibits diffusive behavior. The equilibrated MSD plots for each system can be found in the supplementary material.

The monomer density near the wall is unaffected by the chain length and polymer architecture. The standard liquid-like layering of polymer monomers is observed for the monomers that are close to the wall, as shown in Fig. 2(a). While the ring polymers lack chain ends, this architectural difference between ring and linear polymers has no effect on the monomer-scale packing close to the wall. Similarities of monomer density profiles between linear and ring polymers were also captured by Lyulin et al. in their simulation study.⁴⁹ However, for the center of mass density profiles shown in Fig. 2(b) for films with H = 10σ, there is a pronounced chain length and architecture dependence. For smaller rings with N = 50 and 100, the center of mass density exhibits a peak close to the wall. For larger rings, the density profile flattens for N = 200 and becomes pronounced in the film center for N = 350. Furthermore, unlike the similarity observed for the monomer density profile between the linear and ring polymers, the peak adjacent to the walls of linear polymers is much smaller than that of the ring polymers. Due to the overall more compact nature of ring polymers, packing at the chain scale is significantly perturbed by confinement even though the monomer packing is the same for both linear and ring chains.

The conformation of ring polymer melts under the thin film has been analyzed in terms of radius of gyration. The results for R_g in the direction parallel to the walls are shown as a function of chain lengths under different confined conditions in Fig. 3(a). For bulk systems, the exponent ν in the scales R_g ∼ N^ν of ring polymer melts is ∼0.43, which demonstrates the absence of screening of the excluded volume in the ring melts. This result agrees well with the previously reported exponent ν less than 1/2 (ν ≈ 0.4–0.43 and ν = 1/3 for very long chains) where ring polymers do not conform to “ideal” with screened excluded volume interactions as linear chains.^8,10,16,50 Once ring polymers are confined, we notice that as the system becomes more confined (smaller thickness), the exponent ν of ring polymers starts to approach to that in the strictly two-dimensional (d = 2) melts (ν = 1/2), where the chains adopt compact configurations and become segregated. This result agrees with the findings of Meyer and co-workers regarding the segregations of linear chain melts in two dimensions.⁵⁷ Meanwhile, we provide the data of linear chains from the work of Pressly et al. as comparison.

The normalized radius of gyration of the polymer rings in the directions parallel and perpendicular to the wall is displayed in Fig. 3(b) as a function of the degree of confinement δ = H/2Rg₀, where Rg₀ denotes the radius of gyration in the bulk. The ring chain conformations in the parallel direction are slightly extended with stronger confinement, like the behavior of the linear chains. In the perpendicular direction, the normalized conformation of ring polymers is significantly reduced, and the dependence on δ is strongest below δ < 1. In the range 1 < δ < 2, the ring polymers are slightly less compressed than their linear analogs, but the difference is modest. The overall radius of gyration of both polymer configurations is not significantly affected, except in the region very close to the walls. Both ring polymers and linear polymers start to become more compact in the range 0.5 < δ < 1, and in the extreme confined situation δ < 0.5, we notice that the linear chain is compressed by the walls, which causes the overall radius of gyration to increase.

The center of mass diffusion coefficients (normalized by the linear chain value for N = 50) of ring polymers are plotted as a function of the chain length for different film thicknesses in Fig. 4(a). The diffusivity of the bulk ring system exhibits the scaling behavior of D₀ ∼ N^−1.2, which is consistent with theoretical results for unentangled ring polymers^5,12 and demonstrates the absence of entanglements in the ring polymers in this study. Based on previous reported MD simulations,^50,58 we observe that the shapes of our ring polymers are very close to percolation clusters,⁵⁹ which are used to describe randomly branched polymers with screened excluded volume interactions. This could help explain why the diffusion behaviors of ring melts follow Rouse-like dynamics better than linear polymers of the same chain lengths. We also show the scaling behavior of the entangled bulk linear polymer D₀ ∼ N^−2.15 from the work of Pressly et al. as comparison. The difference of scaling behaviors between linear and ring polymer melts departs from the scaling behavior in solutions,⁶⁰ where it was demonstrated that the scaling exponent υ of D ∼ N^−υ is similar between linear polymers and ring polymers (υ is between 0.62 and 0.69 for both polymers). When the ring polymers are symmetrically confined in a thin film, the in-plane diffusion coefficient D_∥ decreases compared to D₀ as the film thickness decreases. However, the scaling with N is not affected by confinement because there is no disentanglement effect, as observed in linear polymers.^36,38

The normalized in-plane diffusion coefficient under confinement for ring polymers decreases monotonically as the channel thickness narrows (Fig. 4(b)). This reduction of the ring polymer normalized diffusion coefficient reached ∼60% and is independent of the chain length. We posit that for unentangled ring polymers, the diffusion coefficients primarily change due to the interaction between the confining walls and the polymer monomers, which causes a change of the monomeric friction that is independent of the chain length. This result is also consistent with the observation that the monomer-scale packing is unchanged near the walls as a function of chain length. For comparison, Fig. 4(b) includes the normalized in-plane diffusion coefficient of linear entangled polymers⁴³ with N = 200. Unlike unentangled ring polymers, as the channel thickness decreases, a significant increase of the diffusivity of the linear chains compared to the bulk is observed, which is a result of disentanglement induced by confinement.

The correlation hole effect, which is a weak center of mass repulsion in the potential of mean force between polymer chains in a melt, becomes stronger when the molecules are confined. Compared to their linear counterparts, ring polymers are more compact, which amplifies the correlation hole effect. To quantify the correlation hole effect, the polymer self-density ρ_self is calculated by treating the center of mass of a test chain as the origin and calculating the density of monomers of the test chain as a function of distance from its center of mass, as shown in Fig. 5. For each system, ρ_self goes to zero at large distances, as expected. However, the normalized self-concentration of the ring polymer near its center of mass (r = 0) depends on both the ring size and the degree of confinement. For small rings [Fig. 5(a)], the normalized ρ_self is nearly independent of the film thickness with only a slight increase (∼6%) in the most confined system. As the ring size increases to N = 100, Fig. 5(b) shows an overall decrease in the normalized ρ_self relative to N = 50 (as expected) and a more pronounced correlation hole effect for r < 0.5Rg₀ particularly in the thinnest film, H = 5σ. These trends continue for the larger rings [Figs. 5(c) and 5(d)], where the correlation hole effect becomes evident in films of thicknesses 5σ and 7σ, and even 10σ when N = 350.

To quantitatively compare the correlation hole of all ring sizes, we plot the relative change in the self-density at r → 0 as a function of the dimensionless film thickness, $H2Rg0$ (Fig. 6). The value of ρ_self(r → 0) is extrapolated to the origin after fitting a polynomial ρ_self(r) to the curves in Fig. 5 for r < Rg₀. We find that the value of ρ_self(r → 0) relative to the bulk value starts to increase for films thinner than H = 2Rg₀ as the polymers’ conformations are deformed due to the confinement. In the systems with H/2Rg₀ < 1, the data appear to fall onto a master curve with a slope of ∼−1.4, though clearly more data over a broader range of confinement would be needed to verify this scaling.

The Rouse model is a well-known and simple model that has been widely used in both experiments and simulations to describe the chain dynamics for unentangled linear polymers. It has also been successfully applied to describe the dynamics of ring polymers with the slight modification that the number of modes has to be even.^4,5 The vanishing of odd modes of a ring polymer can be explained by the fact that a wave vector that starts from any point (or bead) along the ring should practically complete a full period (a wavelength multiplied by an integer value) when it is halfway through the polymer contour, as required by the symmetry of the two halves of the paths along the ring polymer backbone.⁵ The Rouse modes of a ring polymer of size N are defined as

where r_n(t) is the position of the nth monomer in the chain at time t. Given X_p(t), one can also obtain the normalized time correlation of the Rouse modes as^3,61

The analysis of the Rouse dynamics is valuable because it provides a lucid view of the dynamics of the chains on scales involving N/p monomers for various Rouse modes.

The resulting autocorrelation functions are shown in Fig. 7(a) along with fits to a stretched exponential function $e−(t/τp,fit)βp.$ We then obtain τ_p as the integral of the fitted function, which leads to $τp=τp,fitβpΓ(1βp)$⁠, where Γ(x) is the Gamma function. For bulk systems, as we expected, the dynamic behavior of the ring polymers follows the Rouse mode scaling $τp∼(Np)2$ except for large p values, as shown in Fig. 7(b). The deviations at large p are due to packing effects on short length scales. The consistent scaling of τ_p ∼ p⁻² in the bulk is consistent with our observation above that the rings are not entangled even for the chain length of N = 350.

To analyze the effect of confinement on the dynamics of ring polymers, we compare the relaxation time τ_p of the different modes for chains with N = 50 and N = 350 under varying degrees of confinement in Figs. 7(c) and 7(d), respectively. We have scaled τ_p by the product p²/N² to more easily show the deviations from the expected bulk behavior. A monotonic increase of τ_p at all Rouse modes is observed as the film thickness decreases, which is attributed to the friction between the confining surface and the polymer monomers. This observation applies to all the polymer chain lengths in this work, even for the longest N = 350, Fig. 7(d), demonstrating that the dynamics of these ring polymer systems are dominated by the interfacial interactions between polymers and confinement walls.

The intermediate scattering function F_s(Q, t) is the spatial Fourier transform of the self-part of the Van Hove correlation function G_s(r, t),^31,62 which can be calculated as

where N is the total number of monomers. Since we are interested in the local relaxation behavior of monomers at positions under the confinements, we will analyze |Q| = 7.1σ⁻¹, which approximately corresponds to the first peak of the static structure factor. To study the effect of confinement on the local dynamics of ring polymers, F_s(Q_xy, t) is calculated using wave vectors in the plane of the film, as shown in the supplementary material. The changes in the local relaxation time τ_Qxy for bulk and confined polymers are extracted via the same procedure, as described above for the Rouse mode analysis. We plot τ_Qxy(z) as a function of monomer position within confined thin films with thicknesses of 5σ, 10σ, and 20σ (Fig. 8). The changes in the relaxation time at this elevated temperature extend a few monomer diameters into the films from the interfaces. The largest observed changes at the interfaces are found in the thinnest films and are only a factor of ∼3.5 slower. The slowdown of local dynamics for those unentangled polymers is also independent of the chain length. These findings are consistent with previous studies on linear polymer chains demonstrating that for unentangled systems, the increase of the chain relaxation time is caused by the interactions with the walls.³⁹ 

Using MD simulations, we have examined the structure and dynamics of unknotted ring polymer melts under athermal, symmetric thin film confinement. We have varied the film thickness from 5σ to 20σ for ring polymer chain lengths N = 50, 100, 200, and 350, and the analysis of the diffusion coefficient as a function of N indicates that all chain lengths are within the unentangled regime. In all confined systems, a pronounced chain length dependence is observed in the center of mass density profile. Specifically, at a constant film thickness, a maximum in the center of mass density develops with increasing ring size and is larger than the corresponding linear polymers. We also examine the chain conformation by calculating the radius of gyration in both perpendicular and parallel directions and find that chains are stretched along the parallel direction and compressed perpendicular to the walls, comparable to the linear polymer case. To characterize the correlation hole effect and its enhancement by confinement, we analyze the self-density ρ_self(r → 0) as a function of the degree of confinement $H2Rg0$⁠. With decreasing $H2Rg0$⁠, there is little change in the normalized ρ_self(r → 0) until $H2Rg0<1$ when the excluded volume interactions among rings increase and consequently enhance the correlation hole effect in confined ring polymers. In highly confined systems, the strong correlation hole effect prevents other rings from penetrating, thus trending toward molecular segregation. In our calculations of the diffusion coefficient, we observed a reduction of the diffusivity, which contrasts with what we observed on entangled linear polymers. Furthermore, through analyzing both the Rouse modes and the segmental dynamics, we demonstrate that the slowdown dynamics can be attributed to the friction between polymers and walls.

This work provides detailed behaviors of confined unknotted ring polymers, which are distinct from linear polymers in both dynamic and static properties, and demonstrates a stronger correlation hole effect once under confinements. These findings provide insights for understanding the confinement effect on the polymers with different configurations. Furthermore, we could potentially mix ring and linear polymers together to manipulate the behavior of the confined polymer mixtures.

Another topic of interest regarding ring polymers is the investigation of properties of knotted ring polymers, given the topological constraints induced by the polymer itself facilitate the study on the influence of molecular topology on the polymer properties. The variation of the knot complexity has been demonstrated to have significant effect on the dynamics and conformations of ring polymers since both molecular rigidity and packing are altered.^63,64 Vargas-Lara et al. stated that increasing the topological complexity number of knotting in the low molecular mass ring polymer melts leads to a rapid slowing down of the molecular dynamics and the chains become more rigid with larger persistence length.⁶⁵ Those properties of knotted ring polymer melts could also be perturbed by confinement, and it will be valuable for us to study it further. Furthermore, we notice that the mass scaling (D ∼ N^β) exponent β for self-diffusion coefficient D in polymer melts is highly dependent on temperature. Jeong et al. investigated the diffusion behavior of unentangled linear alkane chains by varying the temperature far above T_g and found that β vary from −1.8 to −2.7 upon cooling.⁶⁶ They attribute this variation of β to the dependence of the enthalpy and entropy of activation on the number of alkane backbone carbon atoms. It would be interesting for us to study this temperature dependence in our confined coarse-grained systems since Hanakata et al. have demonstrated that the activation enthalpy of linear polymers is affected by the confinements,⁶⁷ which will cause the variation of β as a function of temperature different from that in the bulk once confined.

See supplementary material for the in-plane center of mass mean-squared displacement (MSD) (Figs. S1–S4) and plots of the intermediate scattering function $Fs$ (⁠$Qxy,t$⁠) calculated using wave vectors in the plane of the film with $|Q|=7.1σ−1$ (Figs. S5–S6).

Z.T., K.I.W., and R.A.R. jointly conceived of this project and collaborated on the data interpretation and manuscript preparation. Simulations were performed by Z.T. with technical guidance from R.A.R.

Z.T., K.I.W., and R.A.R. acknowledge support from the DOE Office of Science (Grant No. DE-SC0016421). This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract No. DE-AC02-05CH11231, and XSEDE facilities through Award No. TG-DMR150034.

The data that support the findings of this study are available from the corresponding author upon reasonable request.