Ring-linear mixtures of semiflexible rubber bands

During the synthesis of ring polymers, it is quite common to obtain mixtures containing both ring and linear molecules. Recent studies, including experiments and simulations, have shown that even small quantities of linear molecules can significantly influence the shape of ring polymers, ultimately altering their rheological response. To further explore this phenomenon, we investigated blends of semiflexible linear and ring filaments by using disordered assemblies of open and closed rubber bands. We employed x-ray tomography to analyze the structure of these mixtures, focusing on how the length and composition of linear bands influenced the overall mixture. In contrast to the behavior observed in fully-flexible polymers, our findings revealed that increasing the concentration of linear bands could actually decrease the average size of rings within a semiflexible ring-linear mixture. This outcome is attributable to a reduction in inter-ring threading, which naturally occurs as the proportion of rings diminishes. To validate our findings, we conducted molecular dynamics simulations on semiflexible ring-linear polymer mixtures in bulk. These simulations confirmed that our results


Introduction
Polymer rings are normally synthesized from solutions of linear chains that have reactive points at their ends. Using very dilute solutions, the linear chains are allowed to react with themselves and close, forming unconcatenated ring molecules [1]. However, this synthesis approach has two main drawbacks. On the one hand, the necessary very dilute solutions produce small quantities of rings. On the other hand, this process generates mixtures of rings and linear chains (contaminants that have failed to close).
Until recently, it was almost impossible to obtain samples of polymer rings without contaminants. Now, newly developed purification techniques allow for the precise experimental characterization of the structure and dynamics of polymer rings [2][3][4][5][6][7][8] and comparison with theoretical predictions.
While linear polymers are currently well-understood in both structure and dynamics, the properties of ring polymers remain elusive. Rings do not have ends, which means that these molecules cannot reptate like linear chains to renew and relax configurations after external stress [9]. In this sense, rings are rather challenging because almost everything that is known in polymer dynamics is related to the tube and reptation dynamics of linear chains [10]. It is still debated whether tube theories can be applied to understand the relaxation dynamics of rings [11,12].
In the seminal work by Cates and Deutsch, several conjectures about the structure of polymer rings were presented, predicting that linear contaminants would enlarge the ring molecules and diminish inter-ring contacts [13]. This was expected because linear chains were thought to more easily thread the rings, producing an effective expansion of them. Additionally, the increased threading of rings by the linear contaminants (compared to a pure sample of rings) would also increase the viscosity of the sample, diminishing the diffusion coefficient of rings. These early ideas were used to explain the rheology of linear-ring blends obtained in different polymeric systems, including polystyrene [14,15], polydimethylsiloxane [16], and polyethylene oxide [17]. In recent years, several authors have confirmed the predicted threading of rings by linear chains by developing molecular dynamics (MD) simulations of blends of fully-flexible chains [18][19][20][21][22][23][24][25][26]. Such simulations have shown that threading of rings by linear chains is more likely than threading of rings by other rings, as expected.
Therefore, experiments and simulations have demonstrated that linear contaminants not only produce an effective expansion of rings but also increase the viscosity of the entire system, even for a small fraction of contaminants. Notably, melts of polymer rings are less viscous than melts of linear molecules of the same size [4]. This is because fully-flexible rings adopt crumpled-like configurations [27], which are rather compact without significant inter-molecule overlap and threading [28,29]. However, semiflexible ring molecules, which naturally appear in several biological systems such as circular DNA [30][31][32][33][34][35], are rather different. Here, the line tension produces more expanded configurations of rings, allowing considerable inter-ring threading [36][37][38][39][40][41][42].
A few years ago, we proposed utilizing a simple bundle of rubber bands as a model to investigate the structure of semiflexible polymers and other fibrous materials [43], akin to how random close-packed spheres are employed to examine properties of liquids and glassy systems. Using this straightforward experimental model, we demonstrated that threadings predominantly govern the structure of confined assemblies of long semiflexible rings.
More recently, we explored linear rubber bands and found that, similar to linear polymers, the average number of entanglements increased linearly with ribbon length [44]. Additionally, we observed that entanglements were less frequent near the container's surface, where more ends were present, mirroring trapped polymers. This recent study provided the first experimental evidence supporting the visualization of polymer structures using macroscopic, athermal analogues, confirming the initial intuitive insights of polymer physics pioneers.
Recognizing the extensive presence of threadings in semiflexible ring assemblies, we investigate the structural implications of introducing linear contaminants. This study explores this query by employing x-ray tomography to capture detailed images of mixtures composed of ring and linear rubber bands (refer to figure 1). Our examination primarily revolves around detecting the modifications in these mixtures, while adjusting both the length of the bands and the concentration of contaminants. We use a machine learning-oriented segmentation method for breaking down the tomographies, which facilitates a comprehensive analysis of individual band configurations, threadings, and the spatial arrangement of the bands. To validate the universal applicability of our findings, we draw a comparison between results derived from rubber bands and those originating from MD simulations of semiflexible ring-linear polymer melt mixtures.
The organization of this work is as follows: section 2 provides experimental details on the preparation, imaging, and analysis of mixtures of ring and linear rubber bands. This section also presents details on the MD simulations used to model mixtures of semiflexible ring and linear polymers in bulk. Section 3.1 presents the experimental results on the structure of the rubber band mixtures. In section 3.2, we compare these experimental results with the results obtained from the MD simulations. Finally, section 4 summarizes the main conclusions of this work.

Methods
We constructed heterogeneous mixtures of ring and linear rubber bands, methodically alternating between ring and linear bands as we added them one at a time into a cylindrical receptacle with dimensions of R c = 3.25 cm radius and 7.2 cm height. This process was carried out by rotating the container, mirroring the approach used in previous research [43]. Two distinct categories of bands, denoted as A and B, were utilized, each having specific lengths L of L A = 20.4 cm and L B = 24.1 cm, respectively. Each experimental unit comprised a mixture of ring and linear bands of identical type, with linear bands generated by severing the rings.
To derive average structural properties, we evaluated three independent samples for each composition as shown in table 1. As the two type of bands used were of different length, and in order to maintain a constant total packing fraction of ∼ 0.16, the total number of bands N bands in the assemblies had to be adjusted (N A = 130 and N B = 71) [43,44]. Additionally, the number of rings N rings and linear bands N linear in each mixture were adjusted for both band types to generate mixtures with comparable fractions of linear chains (ϕ).
We utilized x-ray tomography with a CT-Rex (35 µm voxel resolution) to image the internal structure of the band mixtures. The x-ray source operated at a voltage of 100 kV and current of 350 µA, as reported in [43]. The tomograms were obtained by rotating the samples through 360 • in 1600 steps. Typically, a conventional 3D tomography in our project contains around 6 × 10 8 voxels arranged in a 3D matrix with typical dimensions of 960 × 960 × 686. Figure 1 showcases a typical x-ray tomography of a ring-linear mixture of bands type B.
After obtaining the tomographies of the band packings, the most challenging aspect is the segmentation of the data, which involves identifying the individual configurations of bands within the entire assembly. In our previous work, we developed a segmentation approach based on watershed and distance transforms [43,44]. However, this method only identified parts of bands, necessitating an iterative algorithm to join the different parts of the same band. Subsequently, we developed a machine learning segmentation approach using convolutional neural networks (CNNs). CNNs are deep learning networks that enable the extraction of characteristic features from images and volumes [45,46], typically through a sequence of convolution, pooling, and up-sampling operations. We utilized the U-Net architecture [47] to segment the tomographies. This architecture adopts a convolution-deconvolution approach and has been successfully employed for segmenting both 2D and 3D volumetric data [48,49].
In our study, we trained separate U-Net networks for each type of band. We obtained the segmentation masks needed to train the networks using the previously-developed watershed segmentation approach [43,44]. To simplify the process, we initially divided the tomographies into subvolumes of 128 3 . We then input the raw data into the trained network to determine the probability of a given voxel being the center of a band. After processing each subvolume, we reassembled the data in the correct order and conducted a connected component analysis to obtain the individual configurations of bands. Notably, this straightforward machine learning approach was effective in identifying both ring and linear bands. Figure 2 exhibits illustrative tomographic reconstructions of ring-linear mixtures derived from the systems scrutinized. The proportion of linear chains, indicated by light blue bands, ascends progressively from left to right.
To ensure the accuracy of the segmented tomographies, we employed the following verification methods: • We checked that the obtained number of ring and linear bands matched the actual number in the container.
• We verified that the length of the segmented bands was correct. Incorrectly segmented bands were easily identifiable. For instance, two adjacent bands that were not properly segmented appeared as a single band with twice the required length. Conversely, a band that was cut (broken) by the segmentation appeared shorter than expected. • Given the limited number of bands, we could visually inspect the segmented bands and compare them to the original structure to determine whether they corresponded to a ring or linear band.
In cases where the machine learning algorithm failed to completely segment an entire band, we applied the previously-developed iterative algorithm to join the different parts of the same band. Overall, the machine learning segmentation approach using the U-Net CNN proved to be much more effective than the previous watershed transform-based approach. We will provide further details regarding this segmentation process in a separate publication.  Subsequently, we utilized the skeleton transformation to fit a chain of beads along the backbone of each band voxel [50], reducing the initial tomography to chains of particles representing each band, similar to those obtained from MD simulations. This image analysis technique facilitated the calculation of several geometrical and topological properties [43].
To assess the generalizability of the findings on rubber band mixtures, we conducted additional MD simulations of mixtures of ring and linear polymers in bulk. The objective of these simulations was to investigate the structural properties and behavior of fully-flexible and semiflexible chains in systems consisting of 300 polymer chains in bulk. To model the chains, we employed the classical bead-spring coarse-grained polymer model [51]. Each system consisted of a mixture of non-concatenated ring polymers and linear chains, where the fraction of linear chains in the mixture was varied in our simulations using the values ϕ = 0, 0.25, 0.5, and 0.75.
In our simulations, we used N monomers to construct each polymer chain, where the diameter of each monomer was σ, and the mass of each monomer was m. To study the effect of chain flexibility, we fixed the chain length at N = 100 for semiflexible polymers and N = 200 for flexible chains. In both systems we built the melt with C = 300 chains.
Adjacent monomers on the chain were connected by finitely extensible nonlinear elastic (fene) bonds, described by the potential: where r is the distance between two adjacent monomers on a chain, K fene = 30k B T/σ 2 is the spring coefficient, and R 0 = 1.5σ represents the extent of the steric inter-monomer interaction.
To introduce bending rigidity to the semiflexible chains, we employed the following bond angle potential between adjacent bonds: where θ is the angle between adjacent monomers, k angle = 10k B T is the bending energy, and θ 0 is the equilibrium value of the angle (θ 0 ≡ π).
To account for excluded volume interactions and prevent monomer overlaps, every pair of monomers interacted via the repulsive non-bonded Weeks-Chandler-Andersen potential: where ε = k B T and H(x) is the Heaviside step-function. We carried out Langevin dynamics simulations of the system at a reduced temperature of k B T/ε = 1.0 with the HOOMD-blue simulation package [52]. The friction coefficient was set at γ = 0.5, and the equations of motion were integrated using the velocity Verlet algorithm with a time-step of ∆t = 0.005σ (m/ε) 1/2 . The mass of each bead m, its diameter σ, and the energy ε were chosen as fundamental units.
To ensure statistical significance, we conducted multiple simulations for each value of ϕ. Each simulation was run for approximately 10 9 time-steps to allow for equilibration of the system. We found that these relaxation times are enough to equilibrate all the mixtures studied. During the simulations, the chains were placed in a cubic box of size L box = 45, with periodic boundary conditions, at a fixed density of ρ = N C L 3 box ∼ 0.32 and packing fraction ∼ 0.17. This allowed us to study the bulk properties of the polymer mixtures with features similar to the rubber bands, and investigate how the fraction of linear chains affected the overall structure of the system.

Experiments: structure of ring-linear mixtures of rubber bands
We initiate our analysis by determining the semiflexibility of ring and linear bands within the mixtures using the metric of persistence length, a quantifier for the rigidity of polymer chains. In this context, we estimate the persistence length of the rubber bands by leveraging the formula [53]: Here, l b denotes the bond distance and θ represents the angle between consecutive bond vectors. It's worth emphasizing that, according to this definition, the persistence length does not inherently form a property of the band but correlates with the band's conformation within the assembly. Commonly, for flexible chains L/L P ≫ 1, for semiflexible chains L/L P ∼ 1, and for rigid rods L/L P ≪ 1. Figure 3 shows the semiflexibility of rings and linear bands, illustrating the variance of L/L P in the blends with the proportion of linear changes. Intriguingly, figure 3 indicates that the semiflexibility of bands remains fairly consistent across the different blends prepared. Both band types examined fall within the semiflexible regime, with rings generally exhibiting increased rigidity due to the additional confinement, which curtails the potential configurations compared to linear chains. It's also noteworthy that the lengthier bands of type B display greater flexibility than type A bands. Figure 4 presents a comparison of the curvature distributions for pure systems of type B, consisting exclusively of either rings (ϕ = 0) or linear bands (ϕ = 1). This comparison clearly shows a narrower range of curvature values for linear bands as opposed to rings. It's also notable that rings exhibit greater curvature than linear bands, indicated by the shift of the peak of the curvature distribution towards higher values. This difference can be intuitively understood: the rings, due to their inherent need to close, exhibit a more compact structure. In contrast, linear bands, free of such a constraint, can adopt more extended conformations, thereby demonstrating less curvature.
To investigate the influence of the linear contaminants on the configurations of rings, we first study if the curvature of bands change in the different mixtures. The curvature (and also torsion) distributions can be calculated by obtaining the Frenet-Serret equations for the tangent, normal, an binormal vectors, at every point of the bands [43,54]. Figure 5 compares the normalized frequencies of the local curvature κ, for rings and linear bands in both systems, for different composition of linear contaminants.
In both systems (A and B) the curvature distribution of linear bands does not change for different compositions. This is because, locally, the linear bands feel similarly neighboring bands, without distinction of they are rings or linear. Similarly, we have recently found that the global structure of dense assemblies of rings and linear bands are rather similar, although linear bands display more extended configurations [43,54].
In addition, we could not observe a change in the curvature distribution of rings (in any of the systems), consistent with an expansion of rings as a consequence of an increase in the concentration of linear contaminants. Note that an effective expansion of rings would imply a shift toward smaller curvature in the  probability distribution function (PDF). For the smallest rings (system A), the curvature distribution of rings remains almost unaffected when increasing the concentration of linear bands. For the largest bands (system B), we can see a small shift in the curvature distribution towards higher curvatures, which should be actually related with a compression of rings, rather than an expansion.
The change in the conformation of rings can also be quantified through the radius of gyration R g defined as: where ⃗ R i and ⃗ R cm are the positions of the i-bead and the center of mass of the ring, respectively. In figure 6, we show the average gyration ratio of rings, for both systems, as a function of concentration of linear bands ϕ. For a better comparison, in this figure the gyration ratio R g is normalized with the gyration ratio of rings in pure melts R 0 g (without any linear contaminant). In agreement with the observed curvature distributions, this figure shows that for the smallest rings, the average gyration ratio of rings does not considerable change with ϕ. For the longest bands, the gyration ratio of rings decreases for large concentration of linear bands, which is in agreement with the results obtained from the curvature distributions.
Thus, the longest rings seem to compress for large concentration of linear bands, which is the opposite that has been observed in simulations of fully-flexible polymers [18][19][20][21][22][23][24][25][26]. For mixtures of fully-flexible rings and linear chains, it was predicted, and later confirmed by simulations, that rings expands due to linear  contaminants. This was because linear chains thread the rings more often, producing an effective expansion of them.
In order to find the reasons for the decrease in size of the rings, here we analyzed the threading of rings by rings and by linear bands, for different concentration of linear bands, ϕ.
The threading of rings by other rings can be studied through a minimal surface analysis. This technique was proposed to study ring threading in simulations of fully-flexible polymer rings [28,29]. The idea is that the minimal surface spanned by a ring's contour (figure 5a) can be used to estimate the number of topological conflicts among overlapping rings through the intersection of the minimal surfaces. We also used a similar idea to obtain the threading of rings by linear chains. Here the rings are also spanned by their minimal surfaces, but now we can check if the minimal surfaces is intersected by linear bands. This is actually quite simple to do, because linear bands can be approximated by chains of atoms connected by bonds. Thus, it can be easily obtained if some of the bonds of the linear chains cross the minimal surface of a given ring. In a recent work, threading or rings by linear chains was studied by using the Gaussian linking number for two closed loops [23]. But in order to evaluate the threading, the linear chains need to be closed in some way. Our approach seems to be simpler to implement. Figure 7 panels (b) and (c) shows the average number of threading of rings by other rings (squares), and by linear chains (circles), as a function of the fraction of linear chains. As expected, for both systems, the threading events of rings by linear chains increases with ϕ, and the number of threading by rings decreases. Thus, similarly to simulations of fully-flexible chains, the threading of rings by linear chains increases with ϕ. However, contrary to simulations, figure 7 panels (b) and (c) show that in this case of semiflexible rings there are many threadings between rings for low fraction of linear contaminants.
But note that some threadings should not be completely relevant for the structure of the assembly. This is the case of rings which do not overlap by a considerable amount (their minimal surfaces barely cross). In order to see how the overlap between rings changes with ϕ, in figure 7(c) we show the averaged total length of threaded segments Lint, normalized by the length of bands, L. For the longest bands of system B (circles) there is almost no threading between rings at high ϕ. On the contrary, for the smallest bands, even for the most diluted rings there is still considerable ring-ring threading Lint/L ⩾ 1. These results show that for the longest rings, the decrease in the radius of gyration at high fraction of linear contaminants is due to a considerable decrease in the inter-ring threading. Although in such cases, the few rings that remain in the mixture are largely threaded by the linear chains, the rings still decrease in size (R g /R 0 g < 1). This should be related to the pressure felt by rings when threaded by other rings. Due to the line tension of the semiflexible rings, we can expect that inter-ring threading produces a considerable expansion of rings. On the contrary, linear chains are more flexible, just because they do not have the constraint to close, such that the induced expansion on the threaded rings should be smaller.

Simulations: structure of ring-linear mixtures of unconfined polymer blends
In the prior section, experimental results indicated that when blending semi-flexible rubber bands, the rings appear to diminish in size in the presence of linear bands. This finding stands in contrast to the behavior posited by Cates and Deutsch [13], and the simulations of fully-flexible linear-ring polymer blends [18][19][20][21][22][23][24][25][26].
To ascertain that this observed ring shrinkage in the presence of linear bands is not an artifact of a constrained [55] or insufficiently relaxed experimental system, we extended our investigation by conducting MD simulations of linear and ring chain blends in bulk, employing both flexible and semi-flexible chains.
The normalized gyration ratio of rings in simulated polymer blends, illustrated in figure 8, distinctly underscores the different behavior of blends composed of flexible (red circles) versus semi-flexible (blue circles) chains. This discrepancy accentuates the unique interplay between the chains in each type of blend.
Previous simulations noted that the introduction of linear chains to a mixture containing flexible chains tends to cause the ring structures to enlarge. This suggests that incorporating linear chains into flexible blends triggers alterations in the overall conformation of the rings, resulting in augmented dimensions.
Conversely, blends comprising semi-flexible chains present contrasting behavior. Upon the addition of linear chains to these mixtures, the ring structures appear to reduce in size. This observation is consistent with our experimental observations using rubber bands, where we detected a similar reduction in ring dimensions in the presence of linear chains. These divergent results underscore the necessity of understanding the distinct responses of ring structures when linear chains are introduced in both flexible and semi-flexible polymer blends. Such insights could broaden our understanding of the interactions between linear and ring chains, ultimately improving our ability to design and manipulate new materials with customized properties.
In figure 9, we present the average threading of rings as the fraction of linear chains increases in both flexible and semi-flexible blends. This figure provides a comprehensive view of threading behavior in polymer mixtures containing these two types of chains. It illustrates the threading of rings by other rings (black squares), by linear chains (red dots), and the total threading of rings (blue triangles) for each blend type, facilitating a thorough comparison of threading behavior across distinct systems.
As anticipated, the number of threadings by linear chains rises with the fraction of linear chains in both systems, emphasizing their direct influence on threading dynamics within the polymer blends.
Remarkably, the total number of threadings on semi-flexible rings diminishes as the proportion of linear chains grows. This unexpected finding is essential for understanding semi-flexible blend behavior and accounts for the contraction in ring size in these mixtures, contrasting with the expansion seen in flexible chain blends.
In the realm of semiflexible polymers, ring-ring threadings serve to keep the rings in relatively expanded configurations. Under such circumstances, we have observed that the Flory exponent, represented as ν, and determined by the equation √ ⟨r 2 i,j ⟩ = A 0 |i − j| ν , tends to be approximately ν ∼ 0.8. As the fraction of linear chains increases, the prevalence of ring-ring threading diminishes, resulting in a dominance of configurations where rings are threaded by linear chains. For the maximum fractions of linear chains, the Flory exponent seems to decrease to ν ∼ 0.65. It's noteworthy that the conformation of rings gradually shifts towards random-coil configurations, albeit still slightly expanded. In our future research, we plan to systematically investigate the relationship between the Flory exponent of rings and the density of the mixture.
Thus, our simulation results confirm that the experimental observations of rubber bands are not artifacts of finite size effects or insufficient relaxations. Rather, they uncover a fundamental property of semi-flexible blends, resulting from the reduction of inter-ring threading as the fraction of linear chains increases.

Conclusions
In this study, we delved into the experimental investigation of mixtures comprising ring and linear polymer-like chains by examining the structure of ring and linear rubber band mixtures through x-ray tomography.
Contrary to observations in simulations of fully-flexible chains, we found that increasing the concentration of linear chains leads to a decrease in ring size. Threading analysis indicates that this outcome arises from a reduction in ring-ring threading as linear chain concentration increases, causing the rings to contract.
This phenomenon is not observed in fully-flexible systems where inter-ring threadings are scarce. However, in semi-flexible ring melts, numerous inter-ring threadings keep the rings in expanded configurations.
To corroborate these experimental findings, we performed MD simulations of ring-linear polymer blends in bulk composed of both flexible and semi-flexible chains. For blends containing flexible chains, we observed the well-established result that rings expand as the linear chain fraction in the mixture increases. Conversely, for semi-flexible chain blends, rings contract with the increasing linear chain fraction due to a decrease in total threading on rings. These simulations support our experimental observations using rubber bands, illustrating that the observed ring behavior stems from the semi-flexibility of the bands or chains, rather than confinement effects or athermal conditions.
In conclusion, our results unveil intriguing behavior in mixtures of ring and linear semi-flexible polymers, which have potential implications for rheological and mechanical properties. The complex interplay between linear chain concentration and ring structure size, governed by threading dynamics, underscores the intricate interactions within these systems. Gaining insight into the underlying mechanisms that dictate the behavior of semi-flexible polymer blends is essential for designing and developing materials with tailored rheological and mechanical properties.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: https:// github.com/garciana/PolymerRings.