Cluster prevalence in concentrated ring-chain mixtures under shear

Semiflexible ring polymers are known to exhibit clustering behavior and form stacks in concentrated solutions. Recently, weak shear was suggested to re-orient these stacks with flow, a phenomenon more easily visible in more concentrated solutions [Liebetreu et al. , ACS Appl. Polym. Mater. , 2020, 2 (8), 3505–3517, DOI: 10.1021/acsapm.0c00522]. In this work, we investigate the impact of mixing linear chains and rings in a similar system under shear, studying clustering in the presence of semiflexible, rod-like chains. We present a correlation between chain monomer fraction and clustering behavior as linear chains take up less space, thus decreasing the system’s effective density and, subsequently, clustering. However, we suggest mixtures with a low chain concentration to maintain or potentially enhance clustering at equilibrium while this effect is destroyed under shear. The mixing of chains and rings may therefore be used to create more strongly organized structures susceptible to reorientation via weak shear. on the system. Varying the chain monomer fraction while keeping the number of monomers constant at three distinct concentrations in the semi-dilute regime, we find that high chain monomer fractions typically


Introduction
In recent years, the importance of polymer topology for dynamical and rheological properties has gained significant importance. Polymers come in a host of different topologies such as linear chains, stars, dendrimers and rings, and the impact of topological differences on polymer dynamics in dilute solution has been studied extensively both in equilibrium [1][2][3][4][5] and a variety of out-ofequilibrium situations [6][7][8] such as shear flow. Special attention has been given to dynamics under spatial constraints, [9][10][11] and it has recently been shown that linear chains and rings move through narrow segmented channels at different radial distances. 12 Spatial constraints imposed via an increase of monomer density have also been studied extensively for melts 13,14 as well as semi-dilute suspensions. 15,16 Polymers of diverse topology often occur naturally. Examples for ring-shaped polymers include plasmids and the kinetoplast from the mitochondria of trypanosome parasites. 17 Synthetically, polymer topology can be controlled through the use of optical tweezers, 18 and double-stranded DNA rings have been manufactured to produce catenanes, [19][20][21][22][23] a group of interlinked rings.
Semiflexible rings in concentrated solutions are known to exhibit a self-organized, amorphous cluster phase in equilibrium. 15,16,24 In this phase, rings form long stacks without net translational and orientational order. On such stacks, rings are strongly correlated to each other as well as to rings on neighboring stacks. These rings are typically of oblate shape, but some of them may also exhibit a prolate shape and thread through other rings in order to reside within a given stack. Dynamically, the cluster phase results in a cluster glass, 25 an arrested form of matter in which the cluster structures are locked into place but individual rings can hop between clusters, similar to individual particles in cluster crystals. 26 In addition to pure ring solutions or melts, mixtures of rings and linear chains have also attracted considerable attention recently, both theoretically and experimentally, due to the rich dynamics they display, and in particular with a focus on the mechanism of threading of the rings by the chain and the ensuing constraint release mechanism. [27][28][29][30][31][32][33] Furthermore, mixtures of long linear chains and polymer rings have been shown to exhibit an increased viscosity as compared to pure chain solutions. [34][35][36] Recently, semiflexible rings have been shown to exhibit pronounced clustering behavior even under weak shear, whereas strong shear destroys clusters. 37 Such sheared clusters have been shown to re-orient due to the imposed shear and form bands aligned with flow independent of consideration or disregard of fully-developed hydrodynamic interactions. Stacks aligned in this manner maintain their orientation even after cessation of shear. This effect is more pronounced in concentrated solutions and more difficult to isolate for semi-dilute systems. In this work, we investigate the effect of mixing semiflexible linear chains and rings on the overall clustering behavior in the system. We employ Molecular Dynamics simulations of semiflexible polymers coupled to a solvent modeled by Multi-Particle Collision Dynamics 38,39 without fully-developed hydrodynamic interactions 40,41 and study the impact of steady shear on the system. Varying the chain monomer fraction while keeping the number of monomers constant at three distinct concentrations in the semi-dilute regime, we find that high chain monomer fractions typically make the system behave similarly as it would for pure ring solutions at lower monomer densities. We link this effect to the effective density and topological state of the system, the former decreasing with increasing chain monomer fraction. However, mixing in only a few chains conversely improves clustering behavior as chains get trapped inside of stacks of rings and help stabilizing these structures. Under shear, the investigated properties progressively lose their dependence on chain stoichiometry, converging towards values dictated by monomer density as _ g grows. Not only could this effect be used to improve clustering behavior in ring solutions, it might simultaneously simplify reorientation of clusters in a particular direction using weak shear.

Numerical model
We perform Molecular Dynamics (MD) simulations to study semiflexible polymers in the bulk under shear. To this end, we employ the Kremer-Grest bead-spring model 42,43 and an additional bending potential to introduce rigidity. 16 The equations of motion are integrated by Velocity-Verlet. 44,45 We use polymer rings of contour length N R = 50 and chains of contour length N C = 25, ensuring that rings and chains, when fully stretched, are of about the same length and still fit inside the simulation box of volume V = (50 Â 50 Â 50)s 3 where s is the length scale of the Lennard-Jones interaction of the Kremer-Grest model. We couple the molecular-dynamics part of the simulation to a solvent modeled by Multi-Particle Collision Dynamics 38,39 (MPCD) with a modification to disable fully-resolved hydrodynamic interactions 40,41 in the interest of computational efficiency. Viscous heating is prevented by employing a Maxwellian celllevel thermostat. 39 Lastly, we employ Lees-Edwards boundary conditions 46 to establish a linear velocity profile corresponding to planar Couette flow such that the expected solvent velocity v s (r) = _ gyx, in which _ g is the imposed shear rate, r = (x,y,z) denotes an arbitrary location, and x, ŷ, and ẑ indicate flow-, gradient-and vorticity directions.
We use the standard set of MPCD parameters 38,39 corresponding to solvent particle mass m = 1, collision angle a = 1301, and collision cell side length a = s. Similarly, we use a well-established set of MD parameters used by Poier et al. 16 to study cluster-glass phases of semiflexible rings, as well as in subsequent works on similar systems. 11,24,37 This choice of parameters corresponds to Weaks-Chandler-Andersen potential parameters e = k B T and s as a unit of length, FENE potential V FENE (r) = ÀekR 0 2 /s 2 ln[1 À (r/R 0 ) 2 ] with spring constant k = 15k B T/e and critical bond extension R 0 = 1.5s, and lastly a bending potential acting on consecutive bonds at an angle f to each other: V bend (f) = ek B (1 À cos(f)) 2 with rigidity constant k B = 30k B T/e. For rings in equilibrium at contour length N R = 50, this model yields a radius of gyration of hR g,0 i E 6.5s and an expected bond length of hl b i = 0.96s. For chains at contour length N C = 25, one obtains hR g,0 i E5.5s. We employ an MD timestep of dt = 0.002[(k B T) À0.5 m 0.5 s], which is sufficient to prevent concatenation.

Simulation details
Because of the vast number of monomers in our system and the use of Lees-Edwards boundary conditions, 46 we employ our own custom code written in C++/CUDA 47 to run efficiently on GPUs. This code has been tested extensively and successfully utilized in previous works. 37,48,49 It combines several optimization methods like parallelized MPCD, 50 particle sorting, 51 Verletand cell lists, 52 and an adaptation of the cell-list algorithm for Lees-Edwards. 53 We set the size of our simulation box to V = (50 Â 50 Â 50)s 3 and investigate a set of different monomer densities r = N/V = {0.364, 0.456, 0.600}s À3 for monomer counts N. For Fig. 2 and 3 and Fig. 6, we have chosen to use a gray color gradient for r = 0.364s À3 , blue color gradient for r = 0.456s À3 and a red color gradient for r = 0.6s À3 . Building upon the effective density employed by Bernabei et al. 15 and Slimani et al., 25 we define an effective densityr = [C R ÁhV R i + C C ÁhV C i]/V where C X corresponds to ring-(R) or chain (C) count and hV X i to the estimated expected volume of the corresponding objects. Note thatr always refers to the effective density as measured at equilibrium. Bernabei et al. 15 took the cubed equilibrium diameter of gyration D g,0 3 as a volume estimate. However, in the case of chains, this would severly overestimate the average proximity and overlap of objects in the system. Therefore, as an estimate for these volumes, we instead use the eigenvalues l a of the where # is the dyadic product and s i is the vector from the object's center-of-mass to the position of particle i) such that hV X i = hl 1 iÁhl 2 iÁhl 3 i. We further define the chain monomer fraction as Q CR = C C N C /N, i.e., the fraction of number of monomers belonging to chains, C C N C , to the total number N of monomers in the system. Accordingly, low values of Q CR indicate a system dominated by rings. The effective equilibrium densitiesr for different values of r and Q CR can be found in Table 1 below. We follow the method set forth by Bernabei et al. 15 and employed as well in subsequent works 16,25 to identify stacks of rings. To this end, we define the director d i of a ring i to be equivalent to the eigenvector corresponding to the smallest eigenvalue of the gyration tensor Ĝ. As such, d i is always perpendicular to the plane in which the open semiflexible ring resides. We can then employ the following criteria for deciding whether any ring i is on the same stack as a ring j: in which r ij is the distance vector between the rings' centers-ofmass, and Dg = 0.1, v 8 = 3.0s and v > = 2.5s are parameters chosen a posteriori to identify stacks in agreement with visual intuition. The results are qualitatively robust with respect to small variations in these parameters. Because this kind of ordering is unique to semiflexible rings, we omit chains from this consideration. We prepare our mixtures by taking a fully equilibrated system of rings in the cluster-glass phase, cutting random rings open, and allowing the system to equilibrate again. The same initial equilibrated pure-ring system was used for preparing all simulations, and the same initial equilibrated ring-chain mixtures were used for varying shear rates _ g at constant chain monomer fraction Q CR to enable highlighting of shear and chain fraction, and maintaining system comparability. We define the Weissenberg number 54 Wi 0 = _ gÁt R,0 , a quantity computed from shear rate _ g and relaxation time t R,0 of a semiflexible ring in dilute solution in equilibrium. We then simulate each system for about t = 3 Â 10 6 [(k B T) À0.5 m 0.5 s], with the exact number varying slightly due to runtime restrictions on the employed GPU cluster. By fitting the autocorrelation function F(t) of the end-to-end vector for chains and the maximum-distance vector between beads i and i + N R /2 for rings with f (t) = a 0 e Àt/t1 + (1 À a 0 )e Àt/t2 , we obtain t R ¼ Ð 1 0 f ðtÞdt. 55 We determine the dilute-equilibrium relaxation times t R,0 D 62 000[(k B T) À0.5 m 0.5 s] for the chains and t R,0 D 46 000[(k B T) À0.5 m 0.5 s] for our rings.

Ring clustering behavior in ring-chain mixtures
Semiflexible rings in concentrated solutions in equilibrium exhibit clustering behavior and self-organize into stacks. 15,16,24,25 Under sufficiently weak shear, these stacks reorient and align with the flow direction. 37 In this work, we demonstrate the changes introduced by mixing chains and rings into the same system. Specifically, we cut open some of the rings, thus keeping the monomer density r constant.
First, we look at the centers-of-mass of the polymers in our system to obtain an overall view of the ensuing phenomenon after imposing shear. In Fig. 1, we show centers-of-mass of rings (red) and chains (blue) for Wi 0 spanning four orders of magnitude and Q CR between 0.0 and 0.5 with r = 0.6s À3 . At equilibrium, the rings in the system look equally as organized into clusters at Q CR = {0.0, 0.1, 0.25}, whereas shorter clusters can be discerned at Q CR = 0.5. As Wi 0 increases, stacks begin to align with the flow. 37 At Wi 0 = 0.04552, we observe regular alignment for Q CR = {0.0, 0.1}, similarly pronounced alignment for Q CR = 0.25 and clusters already breaking for Q CR Z 0.5. At Wi 0 = 0.45524, stacks are fully aligned for Q CR = {0.0, 0.1}, start dissembling for Q CR = 0.25 and are mostly broken for Q CR Z 0.5. As Wi 0 is increased another order of magnitude, very few and only short clusters can be seen independent of Q CR .
Intuitively, we expect that low values of Q CR would yield similar results to a pure-ring solution or even slightly improve clustering. Additionally, it seems that higher values of Q CR might cause clusters to align and eventually also dissemble at comparatively lower Wi 0 . Increasing Q CR past Q CR D 0.25 severely reduces clustering in the system when compared to lower Q CR values.
We can verify and quantify this behavior using the dispersion index DI of the centers-of-mass of our rings and chains as described by Kam et al. 57 and also Liebetreu and Likos; 37 this quantity is shown in Fig. 2a and b. To this end, we introduce a cubic grid of cell length l = 2.5s and measure sample mean m of the number of centers-of-mass per cell. We label the sample standard deviation of this quantity s. The dispersion index DI is then calculated as DI = s 2 /m. Characteristically, a value of DI = 1 indicates a random arrangement of centers-of-mass in accordance with a Poisson distribution. Values of DI o 1 point at more-than-random uniformity, for example grid-like symmetry for all individual centers-of-mass, whereas DI 4 1 signifies clustering. It should be noted that the justification for employing this parameter is mostly a comparative one. In Fig. 2c, we provide the probability p(h c 4 5) for a ring to belong to a stack with more than 5 members, which produces a similar graph. The advantage of DI, however, is that the choice of 5 members for p(h c 4 5) is arbitrary and not necessarily indicative of clustering. Moreover, removing rings from a given system can drastically alter p(h c 4 5), while DI should be more robust under such changes and provide a more robust measure for clustering when altering Q CR . In addition, the dispersion index DI always has a reference value of 1 to indicate clustering or uniformity in the system. Fig. 2 shows the dispersion index DI both for chains (Fig. 2a) and for rings (Fig. 2b). DI for chains is, across shear rates and effective densitiesr, at around DI t 1. A notable exception occurs for the most concentrated solution r = 0.6s À3 at Q CR = {0.1, 0.25}, where chains appear to exhibit some weak clustering behavior following the strong clustering of the rings in that system. Curiously, this exception is also visible in Fig. 2b for the rings, where equilibrium values of DI at Q CR = {0.1, 0.25} exceed DI at Q CR = 0. A similar effect can be discerned at Q CR = 0.1 for semi-concentrated solutions at r = 0.456s À3 , whereas it vanishes for the most dilute concentration investigated, r = 0.364s À3 . Looking at the effective densitiesr, the effect is visible wherer \ 3.0 (compare Table 1), which we will therefore treat as an estimate for a stabilization threshold in this context. This threshold is also discussed later in the context of Fig. 7 for momentary effective densitiesr m , where these transitions are shown to be linked tor m , but also influenced by Q CR and, subsequently, topology.
The weak clustering of the chains could point at chains becoming trapped inside or between clusters of rings -an observation difficult to quantify, but one which agrees well with our visual impression of the system from Fig. 1. It seems plausible that this trapping could potentially stabilize the clusters under shear, protecting them from breaking into smaller clusters even at high shear rates, but we do not observe such an effect. In fact, for increasing shear, the same behavior can be seen for all the investigated systems: DI decreases with shear, and ultimately, DI for rings starts to converge against DI D 1 independent ofr or Q CR . It appears the chains in this work, at least, are either too short or too rigid to effectively tangle the rings and hold them in place. However, small non-zero values of Q CR seem to at least improve clustering behavior in equilibrium, and they do not appear to influence the slope of DI as shear increases, givenr \ 3.0 (compare Table 1).
Finally, we plot the probability p(h c 4 5) for a randomlyselected ring to be part of a stack with a ring count h c 4 5 in Fig. 2c. Unsurprisingly, this probability decreases consistently with increasing shear as clusters start to break apart. Curiously, at r = {0.456, 0.6}s À3 , low values of Q CR cause well-separated graphs at high shear and overlapping graphs near equilibrium, even though there are less rings in the system as Q CR increases.
This provides further evidence that a small amount of chains can enhance cluster formation, or at the very least does not disrupt it. The observed improvement is, however, small, and we cannot confirm without reasonable doubt that the effect is not just the result of some noise. On the other hand, the consistent behavior across different densities r for values of Fig. 1 Cluster phase in chain-ring mixtures. Centers-of-mass of rings (red) and chains (blue) for a variety of different Weissenberg numbers Wi 0 and chain monomer fractions Q CR . The simulation box is shown in blue, and the system is projected onto the flow-gradient plane. The snapshots were created using VMD, 56 Fig. 2c indicate that future investigations of the system at these parameters might provide further evidence for the effect discussed here, and potentially show that clustering is not just equal but stronger under the addition of a few chains at Q CR D 0.1, at least for sufficient overall densities r.
Next, we look at the rheological properties of the system through the polymeric viscosity 58 Z = Às xy /_ g where s xy is the flow-gradient element of the stress tensor as computed from the modified Kramers expression 59 the total force acting on bead i. We have already established from Fig. 2 that the chains in this work do not effectively tangle the rings to keep them on formed clusters as shear increases. Fig. 3 confirms this effect through clear shear-thinning qualitatively independent ofr and Q CR . We suggest that both the rigidity of the chains and their shortness might be detrimental to shear thickening, with some chains getting trapped within the stacks, but then aligning with them or simply capable of escaping a given stack without causing too much friction. We establish a direct correlation between polymeric viscosity Z and monomer density r, highlighted by three distinct bands at high shear in Fig. 3. This also indicates that any topological influence becomes drastically reduced at high shear, and that the main factor for polymeric viscosity in this regime is the monomer density r rather than the effective densityr or the chain monomer fraction Q CR . Across all investigated mixtures, there seems to be a shear regime in which low values of Q CR lead to lower relative viscosities Z when compared to graphs of higher Q CR at equal r. Unfortunately, the effect is difficult to see and more difficult to isolate. We suggest this occurs because, as clusters are broken apart by shear, chains might cause additional friction and align at a different rate than rings. We will revisit this question in Section 3.3.

Stack orientation under shear
We have defined the director d of a ring as its plane normal, computed from the eigenvector of its gyration tensor Ĝ corresponding to its smallest eigenvalue. The clusters formed under  equilibrium are typically oriented randomly. Under shear, those stacks start to align themselves with flow until they eventually break apart. 37 We show quantitative evidence of this phenomenon in Fig. 4, which describes angles hy x i of stack axis to flow, hy ŷ i of stack axis to gradient and hxi of stack axis to individual ring directors d. The stack axis is computed as the main principle component via Ĝ of the set of centers-of-mass of rings on that stack.
An important feature of these angles is that they show the alignment of stacks as shear increases, up to about Wi 0 D 0.5, where we find a minimum for hy x i in Fig. 4a. For Wi 0 \ 0.5, clusters break apart and long aggregates become increasingly unlikely. Another indicator for this behavior is p(h c 4 5) in Fig. 2c, which shows a rapid decline for Wi 0 \ 0.5 across all values ofr and Q CR . If one compares hy ŷ i to hxi in Fig. 4b and c, one can see hxi mimicking the behavior of hy ŷ i shifted to slightly higher values of Wi 0 . We suggest that stacks, which are expected to have a much higher relaxation time than individual rings, respond to shear before individual rings do, thus causing stacks to align even as rings still maintain their equilibrium orientations. Only at higher shear rates, when rings start to align as well, do the clusters break apart.
The observed alignment effect with shear becomes less pronounced for higher values of Q CR across all investigated monomer densities r. We suggest that this is simply due to less and shorter clusters forming under such conditions as chains become increasingly system-dominating, an observation taken from Fig. 2. Such short clusters can also be detected as a side effect of the employed algorithm for detecting these stacks when rings in the system become increasingly aligned in the same way and happen to pass each other. This effect becomes increasingly pronounced at high shear and directly influences the angles as seen here. For high values of Q CR , the system is dominated by chains, and the few clusters which might form will be increasingly short and short-lived, showing this sideeffect even close to equilibrium.
In the context of chains potentially helping the alignment of clusters under shear, no such effect can be discerned here. At Q CR = {0.1, 0.25}, the graphs for r = 0.6s À3 are almost indistinguishable, so we can conclude that the addition of only a few chains does not significantly alter the alignment of stacks under shear. Overall, the alignment here seems to follow similar rules for when chain monomer fractions start affecting the system as we described before from Fig. 2.

Individual chains and rings in the mixture
We visualize some typical configurations of rings (red) and chains (blue) in the aforementioned simulations at Q CR = {0.1, 0.25} in Fig. 5. One can identify the alignment of both rings and chains as shear increases, as well as spot the onset of twisting for rings, which is more pronounced for still higher values of the shear rate. 37 Chains instead exhibit stretching. Tumbling, which occurs for both chains and rings in these simulations, is difficult to visualize from snapshots. It should be noted that here, we can even see a more complete alignment of rings into the flow-vorticity plane for Wi 0 = 0.45524 and Q CR = 0.25 when compared to Q CR = 0.10 at equal Wi 0 , as well as the onset of stack destruction as shown in Fig. 1 and 2. This observation provides further evidence that as rings align with the flow, clusters start breaking. At constant monomer density r, increasing the chain monomer fraction Q CR means creating a more chain-dominated system in which the remaining rings have more space to orient themselves (r will be lower if r is kept constant).
In Fig. 6, we present instructive properties of individual chains and rings. First, we define the radius of gyration hR g i ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l 1 þ l 2 þ l 3 p for eigenvalues l 1 Z l 2 Z l 3 of the gyration tensor Ĝ 1,60 and visualize it in Fig. 6a and d. Its value typically scales proportional to effective densityr. Semiflexible chains exhibit swelling under shear as they stretch and elongate up to a maximum value, then deflate as shear increases further due to tumbling (Fig. 6a). Rings do not exhibit swelling; their rigidity causes them to not stretch enough to balance out a twisting which is entropically favorable to stretching. Instead, rings deflate under shear (Fig. 6d). However, this twisting nevertheless makes them resemble a prolate object, and their form mimicks that of the chains in strong shear.
To quantify this behavior, we define the prolateness S* which inhibits values from À0.25 (a circle) to 2 (a straight line) and describes how ellipsoidal a shape is. Via the eigenvalues l i of Ĝ, S* can be computed in the following fashion: 1,60 and we present the result in Fig. 6b for chains and Fig. 6e for rings. While semiflexible chains resemble a straight line almost independently of shear rate, rings, especially at high monomer densities, twist themselves into a shape that more and more resembles that of the linear chain. This also provides an explanation  for the viscosity in Fig. 3 -at high shear, rings and chains offer little difference in shape, and thus, monomer density becomes the defining factor in that regime. We confirm from Fig. 6c and f that chains align themselves with the flow axis while rings align themselves into the flowvorticity plane, which confirms our observations from Fig. 5. Note that we are measuring angles to an axis, so we are biasing our values away from perfect alignment.
We finally turn our attention to the question of the impact of the effective densityr on clustering behavior, which is summarized in Fig. 7. The main question here is whether the clustering behavior discussed around the findings presented in Fig. 2 is mainly dictated by effective densityr rather than by the topological effects introduced by mixing chains and rings. To this end, we define the momentary effective densityr m , similarly to the effective densityr, with the volume computed from averaged gyration tensor eigenvalues hl i i not at equilibrium, but instead in any given mixture at Wi 0 4 0.
As the shear rate and the chain monomer fraction Q CR increase, the quantityr m decreases accordingly because rings and chains stretch and deform, reducing thereby the volumes of the ellipsoids of inertia hV C i and hV R i. The values scanned by the quantityr m in mixtures across varying Wi 0 , r and Q CR lie in the same range, independently of the details of the mixture stoichiometry. If the topological differences between chains and rings were negligible and instead their different sizes caused the phenomena discussed in this work, then we should see all values of DI and p(h c ) in Fig. 7 collapse onto a single graph, respectively. This is clearly not the case here, therefore topology is important.
There are other possibilities: if the different clustering strengths are tied only to monomer densities r and topological differences had no effect, we should see three distinct graphs each showing slightly different dependence onr m . This could apply here at least in parts of the figure, but it is not an exhaustive explanation. Besides, we have already established that clustering is stronger for high values of r, so this influence is not surprising. Alternatively, if Q CR was the sole driving force behind clustering strength, we should see the same markers at the same values for DI and p(h c ) for varyingr m , respectively. This is not the case here, either.
Similarly, we can get another indication for the decreasing significance of topology for high values of Wi 0 : this is easiest to see in Fig. 7a where we find overlapping same-colored data points at the lowest exhibited values ofr m , meaning that at these shear rates,r m is almost the same independent of Q CR . Here, we see overlapping datapoints only in the high-shear regime, and the same markers are not always found at the same values of DI or p(h c ). It is no surprise that r would be the main driving force behind clustering strengththis has already been established in Fig. 2. However, on top of this, we show here that Q CR has a significant impact of its own on clustering behavior in the system, one that is not only caused by the difference inr m , especially for moderate shear Wi 0 A [10 À1 , 10 1 ]. Again, if that were the case and the topological difference had less impact than the difference in size, we should see the scatterplots collapse onto single graphs such that the quantities shown in Fig. 7 would depend only onr m , which is not the case.
Close to equilibrium, momentary effective densitiesr m are heavily influenced by Q CR as shapes are distinctly different, but at least for Q CR A [0.0, 0.25], the impact on clustering is negligible, confirming our previous observations about mixing in chains and the parameter regimes in which it impacts the system the strongest. Most importantly, we can discern distinct graphs, meaning that even at same initial effective densities, different clustering behavior can be achieved by varying Q CR .

Conclusions
In this work, we have investigated the effect of cutting open semiflexible rings in a concentrated solution to create a chain-ring mixture. We have denoted the chain monomer fraction as Q CR . We find that small values of Q CR do not significantly alter the clustering-and general behavior of the solvent, and values Q CR \ 0.5 make clustering progressively unlikely. Under weak shear, the addition of chains alters neither clustering behavior nor cluster alignment. However, the addition of chains does significantly alter the effective densitiesr observed in the system. For high Weissenberg numbers Wi 0 , topological influences become negligible as both chains and rings become highly elongated and tumble. In this regime, monomer density r takes precedence over Q CR orr.
Initially, chains might get trapped within clusters, but their length is not sufficient to entangle the stacks, and under shear, the stacks break apart using the same mechanism as described before. 37 We find that the addition of a small number of chains into a system of semiflexible rings (low values of Q CR ) does not appear to affect the clustering behavior of the system, neither with nor without shear. Therefore, experimental investigation of such systems should not require polymer blends that only, exclusively, feature rings.
Future work in this direction should investigate especially the regime of Q CR A [0.0, 0.25] and Wi 0 A [10 À3 , 10 0 ] in which chains might aid the system in clustering. Of particular interest will be the variation of rigidity, either in general or only for the chains to increase the overall entanglement in the system. For this subset of shear rates and chain monomer fractions, additional investigations could also alter chain contour length. It is possible that, for chains of the same contour length as the rings, their effect on clustering behavior might be much stronger, and we encourage the tools developed and presented in this work be applied to further, similar investigations.

Conflicts of interest
There are no conflicts to declare.