III. 2D Linear and ring polymer
3.2.2. Results and discussion
Figure 3. 2. 1 depicts a representative chain configuration for the simulated 2D flexible ring PE melt under a quiescent condition. Various topological constraints of the ring polymer, such as the intermolecular entanglement, mutual ring threading, ring knotting, and concatenation, which are thought to be very crucial to determine conformational and dynamical behaviors of the 3D ring polymer melt system were excluded in the 2D or strictly confined ring polymer melt system due to the dimensional limitation. In 2D system, we can clearly elucidate the intrinsic conformational and dynamical characteristics of ring polymer by comparing them with 2D and 3D linear or ring analogues.
As illustrated in Fig. 3. 2. 1, the overall chain conformation of the nearly 2D atomistic unconcatenated and unknotted ring PE melt is very different from that of the corresponding 3D atomistic unconcatenated and unknotted ring PE melt. As expected, the 2D or strictly confined unconcatenated and unknotted rings do not show the typical topological constraints of 3D rings such as the intermolecular entanglement, intermolecular threading, intramolecular knotting, and concatenations in the snapshots [Fig. 3. 2. 1(b)]. Most of the 2D unconcatenated and unknotted ring polymer chain display the extended, interpenetrated double-folded chain conformation with the local long protrusions or loops while the 3D unconcatenated and unknotted ring polymer chain shows the compact, segregated chain conformation in the snapshots. Interestingly, this overall chain conformation of the 2D ring PE chains qualitatively resembles that of 2D linear PE counterparts. In other words, the extended, interpenetrated chain conformations prevail in both 2D linear and ring polymer melts. Despite the intrinsic molecular structural differences between linear and cyclic polymers, it is reasonable to assume that the extended, interpenetrated double-folded chain forms dominate the structure of the chains in extremely confined melts. Due to no free ends structure of the ring chains, the 2D ring chains seem to be difficult to move into free space and can be squeezed by adjacent chains. The ring chain appears to be compressed by topological constraints.76,125 Therefore, the 2D ring polymer chains tend to be more frequently folded into themselves than the corresponding 2D linear polymer chains, resulting in less extended chain conformation than the 2D linear chains. In the snapshots, the 2D ring polymer chains appear to have more folded conformation or local loops along the chain contour than the corresponding 2D linear polymer chains.
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Figure. 3. 2. 1. (a) Schematics of the monolayer confined systems in which the polymer chains do not overlap with each other, and the size of the confined monolayer is sufficient to allow torsional behavior;
and snapshot of a 2D R_600 observed in the direction of the normal vector of the xz-plane. (b) Snapshots of the overall 3D L_400 (left) and R_400 (middle) and a randomly chosen single configuration in the 3D R_400 (right). (c) Snapshots of the overall 2D L_600 (left) and R_600 (middle) and a randomly chosen single configuration in the 2D R_600 (right).
Figure 3. 2. 2(a) contains the global structural variations of the 2D ring polymer such as the mean- square radius of gyration as a function of molecular weight. The simulated 2D ring polymer melt systems follow well the general scaling behavior of structural properties. The
R
g2 of the simulated 2D and 3D linear polymer chain is scaled to Na, conforming the behavior of the ideal Gaussian chain.34
On the other hand, the Rg2 of the 2D ring polymer melts slightly diverges in the long chain length region Rg2 Na0.84 from the ideal Gaussian chain behavior represented by the plateau in the figure.
It is noted that the simulated 3D ring polymer chain also exhibits the same decreasing tendency in the scaling exponent in the inset of Fig. 3. 2. 2(a). This decreasing tendency of the scaling exponent is consistent with the previous experimental and numerical studies of the 3D bulk unconcatenated and unknotted ring polymer melts.76,83,84,93 This trend in the unconcatenated and unknotted 2D ring polymer melts can be explained by the nonconcatenation topological effects as reported in the previous computational studies of the 3D unconcatenated and unknotted ring polymer melts.76
According to the Rouse model for the unentangled ring polymer chain,76,126 the ratio of the mean- square radius of gyration between 3D unentangled linear and ring polymer chains is theoretically computed as g2 g2
Linear Ring =2
R R in equilibrium. This ratio decreases from 2.10 to 1.35 for the 2D unconcatenated and unknotted ring system while it increases from 2.04 to 2.79 for the 3D unconcatenated and unknotted ring system as a function of N. This is expected to be linked to a non- Gaussian shape of the chain in both cases. As the chain length increases, the 3D cyclic polymer chain tends to have more compact structure than the Gaussian ring chain resulting in the increase of the conformational ratio. However, the conformational ratio decreases because the 2D cyclic polymer chain tends to have more extended conformation than the Gaussian ring chain as the chain length increases.
We also calculated the ratio of mean-square radius of gyration between the identical polymer chains in 2D and 3D, g2 g2
2D 3D
R R
. This conformational ratio is turned out to be 2.6 for the ring C400 chain (or 2.84 for the C1000 ring) and 1.26 for the linear C400 chain (or 1.24 for the linear C600 chain).This means that under the same molecular weight and the same geometric constraints, a ring-shaped polymer chain expands more easily than a corresponding linear-shaped polymer chain. This larger expansion ratio of the ring is thought to result from more compact structure of the 3D unconcatenated and unknotted ring chains than the corresponding 3D linear analogues.
Further details on the variation of chain structure are revealed by the probability distribution function (PDF) of the radius of gyration in Fig. 3. 2. 2(b). As can be seen in the figure, the overall PDF of the 3D linear and ring system displays Gaussian statistics, while the overall PDF of the 2D linear and ring system shows non-Gaussian statistics. As with previous study,1 this non-Gaussian conformational behavior of the 2D linear and ring system is expected to be associated with the increased stiffness of the chain by the geometric constraints of the 2D system. In both 2D and 3D systems, the overall shape of the PDF of the unconcatenated and unknotted ring PE system is narrower in width and higher in height compared to the corresponding linear PE system. As with the linear polymer case in previous
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results,117 the conformational behavior of 2D or extremely confined ring polymers system also may not be properly described by the simulation or theories based on the random walk of polymer chains.
Fig. 3. 2. 2. (a) The mean-square of ring radius of gyration Rg2ring (left axis) versus the one-half of the corresponding value for the linear analogue Rg2linear /2 (right axis) on a linear-linear plot. The inset shows the rescaled mean-square chain radius of gyration Rg2/Na of polymer chains as functions on a log–log plot. The circle and triangle symbols represent ring and linear melts, respectively, and the results of 2D and 3D are distinguished by black and green colors, respectively: Black circle, 2D ring PE melts; black triangle, 2D linear PE melts; green circle, 3D ring PE melts; green triangle, 3D linear PE melts. The data for the 3D linear and ring systems have been taken from ref. 76. (b) Probability distribution functions of Rg/ Rg2 1/2 for linear (lines) and ring (circles) PE melts in 2D (black) and 3D (green) systems. The error bars are smaller than the sizes of the symbols.
Figure 3. 2. 3 provides the local structural properties of the 2D unconcatenated and unknotted ring polymer melts. The local structural characteristics of polymer chains can be analyzed by examining the
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mean-square segmental radius of gyration R sg2( )a with respect to the segment length s of atoms along the chain. As shown in Fig. 3. 2. 3(a), note that the dashed lines and horizontal region represent the local scaling behavior of the simple rigid rod and the Gaussian chain, respectively as a function of the chain segment length scale. For 3D and 2D ring polymer melts, the Rg2
( )
sa sa increases to the maximum point, deviating from the simple rigid rod, in short-to-intermediate chain segment length scale and then decreases slightly in the long chain segment length scale where the chain segment length is comparable to its contour chain length. On the short chain segment length scale, the late deviation of the 2D ring polymer chain from the simple rigid rod line implies that it possesses a longer stiff chain segments than the corresponding 3D ring polymer chain. The characteristic chain segment length at which the maximum value appears is also expected to be related to this tendency. A slight decrease in this property following the maximum point is thought to be related to the intrinsic structure of the ring polymer, i.e., the closed-loop topology. It is interesting that 2D linear and ring polymer exhibit similar overall behavior with respect to the chain segment length scale, except for the discrepancy at the ends resulting from differences in intrinsic molecular architecture (free ends for linear polymers vs. no free ends for the ring polymer).To evaluate the intrinsic chain stiffness, we analyzed the persistence length of 2D and 3D linear and ring polymer melts using the bond correlation function of the bond vector in Fig. 3. 2. 3(b). It is well known that the persistence length can be estimated from the fitting of the curve to a simple exponential function.69 However, the overall trend of curves does not follow the simple exponential decay function except for the 3D linear polymer melt. We roughly estimated the persistence length of each system using the initial decay of the curve before cos
=0. The persistence length of each system is estimated to be 6.74 for the 3D linear PE melts, 4.95 for the 3D ring PE melts, 12.65 for the 2D linear PE melts, and 13.2 for the 2D ring PE melts.It should be noted that the same oscillatory behavior of the bond correlation function has been observed in previous numerical and experimental studies of semiflexible linear and ring chains in the confined system.32,34,116,127-129 From the numerical study of a 2D linear chain in circular and rectangular boxes,129 this feature is thought to be related with an unstable double-folded chain conformation under certain conditions where the persistence length is comparable to the confined box dimensions. Very interestingly, from experimental study of the circular DNA chains in 2D constrained system, 2D circular DNA chains trapped in other circular DNA chains display a double-folded conformation and the oscillatory behavior of the bond correlation function.34,116 It is thought that this situation is similar to the situation in which one ring in the melt is trapped among its neighbors. Hence, we can conjecture that based on the similarity of the oscillatory bond correlation function, both linear and ring chains appear to have similar double-folded chain conformations under geometric constraints.
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A single ring chain in the melt should, on average, be confined by a certain free space blocked by the surrounding chains. Because ring chains do not have free chain ends like linear chains, a single ring chain cannot escape as freely as a single linear chain in the space confined by the surrounding chain, and folds itself towards the main contour of the chain in the confined space, forming a double-folded conformation. Therefore, many loops are expected to appear along the contour of the ring polymer chain.
This characteristic loop along the chain contour, created from topological interaction with surrounding chains, is assumed to be related with the minimum point of the bond correlation curve in the Fig. 3. 2.
3(b). The characteristic chain segment length at which a minimum point is located is thought to be linked to the average size of the local loop structure. Based on this fact, we may think that the 2D linear and ring polymer chains have, on average, longer loops along their contour than the 3D ring polymer chain. This is also consistent with the fact that the simulated semi-flexible 2D ring chain segment will have longer size of the loop than the simulated flexible 3D ring chain segment. The slow growth of 2D ring polymer to zero beyond the minimum point compared to 2D linear polymer chain is supposed to be associated with the intrinsic closed-loop structure of the rings that construct the global minimum point for ring polymer. The deeper minimum points for 2D linear and ring polymer chains compared to 3D ring polymer chains imply that characteristic loops are more dominant in 2D. The 3D linear polymer melt exhibits the simple exponential decay function because topological interactions with surrounding chains such as blocking by neighboring chains are relieved by the motion of free chain ends, resulting in less loop structures.
The contact probability that characterizes the internal structure of the chain is reported in Fig. 3. 2.
3(c). It is noted that the contact probability is the probability that two monomers a distance s apart along the chain meet in space. The cutoff distance defining the contact was set to
2
1/6
. In the figure, initial response of the contact probability, a rapid decrease from a plateau region, in the short chain segment length scale is attributed to the strong spatial correlation imposed by the intramolecular bonding interactions such as bond torsional interactions.The qualitative overall behavior of 2D ring polymer is very similar to that of 2D linear polymer due to increase of chain stiffness in extremely confined system. In particular, slower appearance of the local maximum point for 2D linear and ring polymer systems compared to 3D linear and ring polymer systems support the increase of chain stiffness under geometric constraints. It is noted that the higher contact probability of the 3D ring polymer chains compared to the 3D linear polymer chains from the minimum point to the end is ascribed to the collapsed and compact structure of the ring.
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Figure 3. 2. 3. (a) Rescaled average segmental radius of gyration and (b) bond correlation function between monomers n and m = n + sa separated by sa bonds for linear (lines) and ring (circles) PE melts in 2D (black) and 3D (green) systems. The Roman numerals indicate the characteristic correlation lengths corresponding to where the bond correlation function is (I) equal to zero; (II) minimized; and (III) almost decorrelated for 2D ring melts. (c) Probability of contact between monomers n and m = n + sa separated by sa bonds with respect to the number of bonds along the chain contour for linear (lines) and ring (circles) PE melts in 2D (black) and 3D (green) systems.
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For further internal structure and packing information, we investigated various pair correlation function such as intermolecular and intramolecular pair correlation functions for simulated 2D and 3D confined melts in Fig. 3. 2. 4. There is the strong correlation hole effect for 3D ring polymer melt compared to 3D linear polymer melt because of the collapsed structure of the ring polymers that prevents other chains from penetrating into their territory.76,94,108,130 The strong correlation hole effect is also observed for 2D ring polymer melt compared to 2D linear polymer melt. The 2D surrounding chains cannot penetrate into the extended 2D ring polymer chains (or inner space of the 2D ring polymer) due to the closure structure, resulting in strong correlation hole effect of the 2D ring polymer melt compared to the 2D linear polymer melt. As shown in Fig. 3. 2 .1(b), a 2D ring chain conformation with a large inner surface area implicitly represents a less penetrated conformation. The difference in the magnitude of the characteristics peaks between 2D and 3D ring melt systems in the ginter(r) may be related to the existence of topological constraints in 3D, such as intermolecular entanglement, intermolecular ring threading, etc. In consistent with results of ginter(r), it is reasonable to interpret that the difference in gintra(r) between the linear and ring polymer melts is attributed to the less interpenetration and more compact structure of the 2D ring chains compared to the 2D linear chains for the 2D confined system, and the collapsed structure of the 3D ring polymer chain for the 3D bulk system.
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Figure 3. 2. 4. (a) Intermolecular pair distribution function ginter(r) and (b) intramolecular pair distribution function gintra(r) for linear (lines) and ring (circles) PE melts in 2D (black) and 3D (green) systems.
To investigate the dynamical features of the nearly 2D confined ring polymer system, we examine three types of mean-square displacement (MSD) with respect to time in Fig. 3. 2. 5(a). As can be seen in the figure, the 2D linear and ring polymer systems, independent of topological constraints such as interchain entanglement, ring threading, and concatenation, exhibited very surprising similar dynamical characteristics. The plots of all three types of MSD between 2D linear and ring polymer chains match to each other over all range of time. This clearly demonstrates that, despite differences in intrinsic molecular architecture (free chain end vs. no free chain end), linear and ring polymer chains confined in nearly 2D system move in very similar way. This also implies that there is no significant free end effect of the chain on the dynamical properties of the 2D or extremely confined system. Therefore, as with 2D linear chains,117 ring chains tend to move as a whole with their monomer in concerted manner in extremely confined system. It is noted that for 2D ring polymer, three types of MSD appear very similar trend regardless of the monomer position along the chain. As with 2D and 3D linear chains,1 we
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find that ring polymers in 2D system move faster than 3D ring polymers from the comparison of the g1(t) values at short-to-intermediate time scales. Based on the earlier cross point between g2(t) and g3(t) in the 2D rings than 3D rings, we also confirm that ring chains are inclined to move as a whole with their monomer in concerted manner in extremely confined system.
Figure 3. 2. 5(b) shows atomistic snapshots of a randomly chosen single chain in the narrowly confined 2D L_600 and R_600 system. Similar to 2D atomistic linear PE melt system, 2D ring chains also tend to diffuse as a whole through largely correlated intrasegmental movements, sometimes with repeated stretched and coiled conformations, despite the intrinsic molecular structural differences between linear and cyclic polymers (free chain end vs. no free chain end).
Figure 3. 2. 5. (a) Various mean-square displacements (MSDs) for 2D L_600 (solid lines) and R_600 (dotted lines) systems (left) and the MSDs for 3D L_600 (solid lines) and R_600 (dotted lines) systems (right): the MSD averaged over all the monomers of a chain [g1(t); black], the MSD of monomers with respect to the center of mass of the chain [g2(t); dark green], and the MSD of the center of mass of the chain [g3(t); orange]. (b) Snapshots showing chain movement over time for the 2D L_600 and R_600 system.
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Figure 3. 2. 6(a) displays the diffusion coefficients for the center-of-mass with respect to the chain length. Although 2D ring polymer melts approximately follow a single Rouse-like scaling behavior
1
DG N− , the scaling exponent of 3D ring diverges slightly from the Rouse model DG N−1.3. This difference in scaling exponent between 2D and 3D ring melt system may be attributed to the existence of topological constraints in 3D. As is well known, topological constraints such as interchain entanglement, interchain ring threading, and concatenation, tend to slow the dynamical motion of the chain. Compared to the scaling exponent variation from -0.96 to -2.31 between 2D and 3D linear chains, the scaling variation from -0.93 to -1.3 between 2D and 3D ring chains is smaller. This may be related to the weaker topological effects in ring melt system than linear melt system. It is reasonable to think that qualitatively, unconcatenated and unknotted 2D and 3D ring polymers up to each C1000 and C400
tend to follow the dynamics of the unentangled Rouse chain.
Figure 3. 2. 6(b) provides the relaxation time for the simulated 2D and 3D melts as a function of chain length. The scaling exponent of the relaxation time for simulated 2D and 3D ring melt system approximately agrees with Rouse-like scaling behavior