Molecular Dynamics Simulation of Crystallization Cyclic Polymer Melts

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Molecular Dynamics Simulation of Crystallization Cyclic Polymer Melts As Compared to Their Linear Counterparts Hongyi Xiao,†,§ Chuanfu Luo,*,‡ Dadong Yan,*,† and Jens-Uwe Sommer*,§ †

Department of Physics, Beijing Normal University, 100875 Beijing, China State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, 130022 Changchun, China § Institute Theory of Polymers, Leibniz-Institute of Polymer Research Dresden, 01069 Dresden, Germany ‡

ABSTRACT: Large scale molecular dynamics simulations have been performed in the framework of a coarse-grained poly(vinyl alcohol) model to study the crystallization behavior of long unknotted and nonconcatenated cyclic polymer chains from the melt state. The results are compared with those for chemically identical linear chains. The crystallization and melting points, stem length, crystallinity, and latent heat of melting/crystallization of cyclic polymers are found to be substantially higher than their linear counterpart subjected to the same thermal history. The reduced amount of entanglements of cyclic polymers is suggested to explain the difference between cyclic polymers and their linear analogue. We applied primitive path analysis to quantify the entanglement state for all systems during crystallization and heating. While for linear chains the entanglement length is increasing during crystallization and annealing indicating a partial disentanglement process, the entanglement length is monotonously decreasing after the onset crystallization for cyclic polymers. We suggest that segments which are trapped by the formation of lamellar crystals essentially contribute to the entanglement density in crystallizing cyclic polymers. The increase of stem length (reorganization) during heating, as found in linear system, is not shown in cyclic polymers, in agreement with recent experimental observations [Zardalidis et al. Soft Matter 2016, 12, 8124]. This correlates with the observation of creation of entanglement constraints during crystallization in our simulations.

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INTRODUCTION Cyclic polymers (C-Ps) differ with their linear analogues by a unique feature: The former conserve topology. While linear chains can take any conformation that is available under the local constraint of excluded volume only, cyclic polymers can never change their state of concatenation and thus provide an additional and global constraint for the conformation statistics. If a polymer melt is prepared by concentrating a diluted solution of nonconcatenated C-Ps, the conformational entropy is reduced as compared to linear chains since any concatenated conformation is excluded. More compact conformations of the C-Ps are predicted as a consequence.2−4 In addition, the dynamics in such melts of C-Ps are accelerated if the molar mass of the polymers is larger than the entanglement molecular mass of their linear counterparts. This can be related to the absence of classical reptation dynamics.5−9 Since C-Ps do not differ in any other chemical detail from their linear counterparts, they provide an interesting system to study the impact of topology on polymer crystallization. Crystallization of long polymer chains is known to result in out-of-equilibrium structuresso-called chain-folded lamellar crystals. The origin of the folded chain structures is still an unsolved problem in polymer science since their discovery about 60 years ago.10,11 When rationalizing the appearance and © XXXX American Chemical Society

the properties of chain-folded crystals, two aspects play an important role: First, the equilibrium thermodynamics defining the equilibrium crystallization temperature

Tm0 = ΔH /ΔS

(1)

where ΔH denotes the latent heat of isobaric melting and ΔS = Sm − Sc is the difference in entropy between the melt and the equilibrium crystalline phase. The equilibrium melting temperature, however, is virtually inaccessible in experiments for long chains where folded-chain structures arise. Formally, the equilibrium melting temperature defines the coexistence of an equilibrium crystal phase and its melt. Here, the equilibrium crystal phase is defined by minimizing all constraints such as chain folding and can therefore be associated with the fully extended chain crystal. For this reason the entropy of the equilibrium chain crystal, Sc, should not be influenced by the ring topology in the case of long polymers. By contrast, the entropy of the melt, Sm, should be decreased due to topological constraints in the cyclic melt as mentioned above. As a consequence, according to eq 1, the equilibrium melting Received: July 23, 2017 Revised: November 19, 2017

A

DOI: 10.1021/acs.macromol.7b01570 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules temperature is increased for C-Ps.12 Given a temperature, Tc, at which the crystallization process is observed, the supercooling is defined as ΔT = T0m − Tc. In classical kinetic approaches to polymer crystallization the supercooling is the key quantity which controls the lamellar thickness and thus the degree of chain folding.13−15 On the basis of this simple thermodynamic argument, one expects a higher degree of supercooling under the same thermal history, i.e., at fixed Tc, for C-Ps melts as compared to linear chains.12 This is a priori related to thinner lamellae and a higher degree of folding according to classical nucleation models if other quantities of the models are not changed. Second, entanglements have been considered to have a direct influence on the crystallization process although a quantitative relation between the degree of entanglement and properties of the crystalline polymers is not known yet. Experimental evidence comes from polymer melts where the entanglements are partially depleted by preparation. Here, it has been demonstrated that significantly higher melting temperatures can be observed for crystals formed under such conditions.16,17 Entanglements in the melt raise the nucleation barrier during crystallization. Our own studies using computer simulations have demonstrated that the entanglement length, as obtained by primitive path methods, seems to determine the lamellar thickness for linear chains directly.18 Variation of various parameters that influence the entanglement length such as concentration, surface effects, and thermal history supported this result.19−21 Thus, the reduction of the degree of entanglements in C-Ps should lead to the opposing effect to what is expected from the thermodynamic−kinetic argument and its application to nucleation models. Therefore, the direct comparison of the crystallization behavior of C-Ps and their linear analogues is expected to shed some light on the validity of kinetic approaches to polymer crystallization, too. Some experimental results have been accumulated for the crystallization of cyclic polymers in comparison with linear chains; however, partially controversial conclusion have been drawn.12,22−35 A recent review comparing various experimental studies can be found in ref 22. A broad consensus is obtained about the fact that the crystalline structure is not influenced by the ring topology which strengthens the assumption of the same equilibrium heat of fusion, ΔH, irrespectively of the ring topology. Cyclic polymers investigated so far are usually found to have higher melting temperature, Tm12,25,29,30,34−37 (not to confuse with the equilibrium melting point, T0m), and higher crystallinity34 than their linear counterparts under a given thermal history. With respect to the lamellar thickness opposing trends have been reported (see ref 22 for comparison). However, a crucial aspect of the experiments, particularly in the melt state, is the purification of the C-Ps. A small amount of linear chains can cause a qualitative effect on the state of entanglement and on the dynamics of C−P-melts,9 a fact which might explain some of the contradictory results.22 In the present study we apply computer simulations to compare the crystallization behavior between linear chains and C−P-melts. As compared to the experiments any contamination with linear chains can be avoided, and pure nonconcatenated and unknotted cyclic polymers can be investigated. In previous work using molecular dynamics simulations of a coarse-grained polymer model38,39 we were able to obtain some quantitative relations between the entanglement length and the lamellar thickness. In particular, we have shown that the entanglement length as obtained by primitive path methods

at the given state of the melt and in situ seems to select the lamellar thickness without referring to any kinetic model. For instance, we could explain the reduction of the crystalline stem length by decreasing the crystallization temperature by the reduction of the average entanglement length due to stiffening of the chain contours.40 Local entanglement length is found to be correlated with both crystallinity and crystalline stem length at the same locus.18 Increasing the entanglement length by diluting the melt of long chains with very short chains again leads to an increasing lamellar thickness.19 The increase of entanglement length by the proximity of a flat wall can explain the nonspecific nucleation and preferred growth of polymers at substrates.20 Cyclic polymers provide a unique system to change the entanglement length as compared to linear chains but leaving all other parameters of the polymers invariant. While it seems obvious that entanglements are depleted in melts of nonconcatenated C-Ps, the viewpoint that entanglements are completely absent for long C-Ps is naive.1,9 Using primitive path analysis, MD-simulations have shown that the entanglement length is increased in melts of long C-Ps but is smaller that the half-extension length of the polymers.3,7 However, the meaning of the entanglement length in C-Ps remains an open topic since local definitions such as the packing length arguments seem not much influenced by the ring topology either. Our simulations show a substantial increase in the crystallization and melting temperatures combined with higher crystallinity for C-Ps as compared with the linear analogue. Together with an increase of the stem length, these results support the view that entanglements control polymer crystallization. The particular role of entanglements dynamically created by monomers fixed in the crystalline phase of CPs is discussed.

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MODEL AND METHODS The coarse-grained poly(vinyl alcohol) (CG-PVA) model developed by Meyer and Müller-Plathe is used.41−44 Molecular dynamics (MD) simulations using the periodic boundary condition and NPT ensemble at 1 atm are carried out with a patched LAMMPS code.39,45 Three cyclic polymer systems with same total size (106 monomers) and various chain lengths are studied in this work, denoted as C1, C2, and C3. For comparison, the same cooling−heating cycle is also carried out to one linear polymer system, denoted as L. For more details about the results for linear chains we also refer the reader to our previous work. The specific parameters of C1, C2, C3, and L are listed in Table 1. The initial conformations of cyclic systems are relaxed over 3 × 107 time steps at T = 1.0 (550 K), which is far above the melting point, while the initial conformation of the linear system is relaxed for more than 8 × 107 time steps. The relaxation time of cyclic polymer is significantly shorter than that of linear system because the equilibrium entangleTable 1. Definition of the Studied Systemsa abbrev C1 C2 C3 L

component cyclic cyclic cyclic linear

chains chains chains chains

chain length

chain number

500 1000 2000 1000

2000 1000 500 1000

a Cyclic systems with three different lengths and one linear melt are simulated.

B


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we denote its value by N0e in this work. We will discuss the relation between N0e and Ne in detail in the Entanglements section.

ment state of cyclic polymers is diluted. We had tested relaxation time as long as 108 time steps for smaller systems, and there is no notable decrease of entanglement length. Starting from unknotted and nonconcatenated conformations the topological state is preserved by avoiding cyclic chains and segments from penetrating each other. Continuous cooling (from T = 1.0 to T = 0.6) which was directly followed by heating (from T = 0.6 to T = 0.93) is performed in 107 MD steps (∼350 ns). The time step of MD integration was 0.01 (∼35 fs). After the trajectories of the MD simulations are obtained, a revised method of primitive path analysis (PPA) first proposed by Everaers et al.8 is applied to analyze the entanglement states during various stages of crystallization and melting. For linear polymers, the basic idea of PPA is to remove intrachain repulsion and let the polymer chains shrink simultaneously while keeping their ends in fixed positions. This procedure results in so-called primitive paths consisting of kinks and straights formed by sequences of monomers. We use the directly measured number of monomers in a straight (defined by bending angles θ > 170°) between two adjacent kinks in a primitive path to define the entanglement length (Ne).40 All monomers in the same straight share the same value of entanglement length, denoted as Ne,i(T) for the ith monomer at temperature T. The weight-averaged value is given by Ne(T) = ∑M i=1Ne,i(T)/M, where M is the total number of monomers in the system. The PPA method has been proven to be reliable for quantitative estimations of Ne when it is compared with several revised methods,8,46,47 such as Larson’s length minimization scheme,48 Kröger’s Z algorithm,49 and Theodorou’s CReTA algorithm.50 In our recent work19 we studied the entanglement effect during the crystallization of concentrated solutions by using both the PPA and the Z1 method. We found that both methods yield the same behavior for the entanglement state during crystallization, annealing, and melting. Even with respect to the absolute value of the entanglement length the difference between PPA and Z1 is decreased for a decreasing entanglement density such as by diluting the polymers. We carried out some test runs using the Z1 method which have confirmed this picture. We note that we were interested here only in the average values of the entanglement length and not in more subtle aspects of the tube and the primitive path.51 The absence of chain ends in cyclic polymers not only suppresses the typical reptation dynamics common to linear and branched chains but also makes it necessary to modify the primitive path method. We follow the procedure of ref 3, and we fix two randomly selected points denoted as “pseudo-ends” which divide a single ring-polymer into two parts of equal length. The PPA is then implemented under these constraints. At least two points are necessary to be fixed for carrying out the PPA to prevent a trivial collapse of all chains. This can be directly seen from the sketch in Figure 8. To indicate the justification of the arbitrary selection of pseudoends, different pairs of pseudoends are tried. We found that changing the pseudoends has little influence on the of kinks’ location in the primitive path. The pinning of two monomers in the primitive path analysis of C-Ps natually provides a numbering scheme for the monomers: In general, we call the monomer fixed by one pseudoend the monomer “0”. For linear chains monomer counting starts as usual from one of the ends. To distinguish the entanglement length in cyclic polymer as obtained by the method of pseudoends from the one obtained for linear chains,

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RESULTS AND DISCUSSION Thermodynamic Properties. At T = 1.0, the CG-PVA polymers are in the melt state, and polymer chains take random-coil conformations as shown in the left panel of Figure 1. After cooling to T = 0.6, a semicrystalline state is observed, as

Figure 1. Section view of the snapshots of the cyclic polymer C2 at T = 1.0 (left) and T = 0.6 (right) plotted using the atomistic configuration viewer Atomeye.52 Here different colors denote different chains. At T = 0.6, most chains form lamellae, although some amorphous regions remain.

displayed in the right panel of Figure 1. We will call this for simplicity the crystalline phase in the following. In the crystalline phase, most polymer chains are locally stretched, aligned, and folded to form lamellar crystals. Amorphous segments are excluded to the interfaces between lamellae and form amorphous regions. The top-right corner (and some other regions) of the right panel in Figure 1 demonstrates the hexagonal crystalline structure as expected from the coarsegrained model. A closer analysis of the crystalline domains display lateral dimension of lamellae which are up to 5 times larger than the average stem length (crystalline thickness). Our cooling protocol allows for homogeneous nucleation which causes many crystalline domains formed at the same time. Some of them are stacked compactly leading to multiple lamellae with the same direction of stems. Folding motifs of individual chains display tight folds dominating to the interlamellar phase. In Figure 2, we display the specific volume (v) as a function of temperature during the cooling−heating cycle. All graphs show the typical hysteresis which is characteristic for semicrystalline polymers and indicates the nonequilibrium character of the crystalline phase. From the inflection points of v(T), we can estimate the apparent crystallization temperatures and the corresponding melting temperatures. For the ring polymers C1, C2, and C3 we obtain Tc = 0.767, 0.763, and 0.772 and Tm = 0.900, 0.903, and 0.913, respectively. Both temperatures are significantly higher than those for the linear polymer: TLc = 0.739 and TLm = 0.882. The increase of the crystallization/ melting temperature in C2, which has the same degree of polymerization as the linear melt, is about 13.2/11.55 K. These values are higher but comparable to experimental results on cyclic PCL given in ref 22 where values of the order of 4−6 K have been reported. In experiments, specific heat at constant pressure can be measured by calorimetry, and the apparent nonequilibrium crystallization and melting temperatures are often determined C


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Figure 2. Specific volume (v) versus temperature (T) during the cooling/heating cycle. The blue and red lines represent the cooling and heating process, respectively. The drop/jump of specific volume indicates phase transitions. The vertical dashed/dotted lines mark the crystallization/melting points of the cyclic polymer C1. It is obvious that both crystallization and melting points of cyclic systems are higher than those of linear system.

Figure 3. Specific heat, cp, as a function of temperature (T) during the cooling/heating cycle. Blue and red lines represent the time evolution in cooling/heating process, respectively. Two peaks in the cp−T curve mark the occurrence of crystallization and melting. The vertical dashed/dotted lines dash indicate the crystallization/melting point of the cyclic polymer C1.

value dc = 14, which corresponds to 28 successive monomers, and the bond angular range defining the tt-state, are obtained from histogram analysis of dtt in the melt and in the crystalline state.43,44 The apparent crystallization and melting points as estimated by the inflection points in the left panel in Figure 4 are Tc = 0.772, 0.768, 0.776, and 0.744 and Tm = 0.9026, 0.906, 0.9094, and 0.8788 for all systems from top to bottom, respectively. They agree with the results obtained by the other methods described above. Our simulations show that the average stem lengths formed by cyclic polymers in the crystalline state are up to 50% larger than that of linear chains. The average over the distribution of longer stems (dtt > dc) is about 10% larger for cyclic polymers. During heating, crystal thickening can be observed only for the linear system and only if the threshold value dc is considered. Such annealing behavior is typical for the nonequilibrium state of polymer crystals and is usually related to the sliding motion of stems within the crystal. It is interesting to note that crystal thickening is not observed for cyclic chains, in agreement with recent experimental observations.1 This will be later discussed in the context of entanglements. In Figure 5 we show the crystallinity during the cooling− heating processes. The crystallinity is defined by the number fraction of monomers in crystalline stems, c = Mc/M, where Mc is the number of monomers forming stems and M is the total number of monomers in the system. The crystallinity indicates the same crystallization/melting temperatures as reported above for the other observables. The crystallinity of cyclic polymers is significantly higher (up to 100%) than that of linear polymers under the same thermal history. Single Chain Properties. As we have noted above, the chain lengths used in our simulations lead to folded chain crystals. In Figure 6 snapshots of three typical single chains

by the extothermic/endothermic crystallization/melting peaks of the heat flux. The heat flux dq per monomer unit in MD simulations can be calculated by the first law of thermodynamics as dq = du + p dv, where the changes of energy du and volume dv (also per monomer unit) can be directly calculated from simulations. The isobaric specific heat cp is defined by

⎛ dq ⎞ cp = ⎜ ⎟ ⎝ dT ⎠ p The results for cp is given in Figure 3. The apparent crystallization and melting points given by the peaks of the specific heat are Tc = 0.769, 0.764, 0.771, and 0.741 and Tm = 0.908, 0.909, 0.919, and 0.887 for C1, C2, C3, and L, respectively, which are consistent with the values obtained from the inflection points of the v−T curves in Figure 2. As the chain length of the C-Ps is increasing, Tc stays nearly constant, but Tm shifts systematically toward higher temperatures. Properties of the Crystalline State. The lamellar crystals consist of parallel stems connected by loops or tight folds within the chains; see also Figure 6 for typical snapshots of individual chains. The stem length, dtt, is calculated by the length of a sequence comprising successive trans−trans (tt) states within the chain. The lamellar thickness can be defined by the average length of stems. Alternatively, we can consider stems longer than a threshold value dc. The threshold, dc, is introduced to disregard the natural occurrence of partially extended sequences due to finite rigidity in the melt state. In the left panel of Figure 4 we display the results for the average stem length without cutoff length, dc = 0. The right panel displays the result for dc = 14. The trans−trans state is defined by a bond angle larger than 150° between two successive monomer bonds. The threshold D


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Figure 4. Average stem length during cooling/heating cycle. The left panel displays the stem length averaging for all monomers in systems (dtt). The vertical dashed/dotted lines dash mark the crystallization/melting point of the cyclic polymer C1. The right panel displays stem length averaging for only the monomers whose stem length is longer than 14 (dttc). Blue and red lines represent the cooling and heating process, respectively.

loops formed by two successive stems in the same chain, we found a strong increase of short loops for cyclic polymers as compared with linear chains. Dangling loops, or distant re-entry behavior, are rare. The prominent occurrence of hairpin folds which participate in crystal domains of different orientations resembles the lattice-tree model of entangled cyclic polymers as proposed by Rubinstein.9,53 In this model the cyclic chain is assumed to form a double-stranded hyperbranched structures to maximizes conformational entropy in the presence of topological obstacles (see Figure 8). Such a conformation would naturally provide hairpins as crystallization motifs and tend to preserve the length scale of the topological constraints as discussed below. To analyze the single chain conformations, we investigate the *, radius of gyration, Rg, and the effective persistence length, C∞ of the individual chains. Here, C∞ * is defined by C∞ * = (1 + ⟨cos θ⟩)/(1 − ⟨cos θ⟩), where θ is the angle between two adjacent bond vectors and ⟨...⟩ denotes the average over all bonds. We note that C*∞ should be only related to the persistence length in the disordered state, while it is clearly correlated with the stem length in the crystalline phase. The results are displayed in Figure 7. In the melt state of C-Ps we found Rg(N) ∼ Nν, with ν varies from 1/3 to 2/5 by using a larger range of chain length not shown in this work. This agrees with theoretical predictions4,54 and previous simulations.3,6,7 At the crystallization point, both C∞ * and Rg increase rapidly, which is related to the formation of stems. When temperature decreases to 0.6, C*∞ for cyclic polymers reach a value higher than for linear chains. This is in full accord with the behavior obtained for the stem length. During the subsequent heating, Rg in each system increases continuously until the melting point. This indicates again that the polymer crystal is far from thermodynamic equilibrium. Obviously, the increase of Rg during heating cannot be simply related to the stem length

Figure 5. Crystallinity during cooling/heating cycle. Blue and red lines represent the cooling/heating process, respectively.

conformations taken from C2 are displayed. The beads with warm colors represent stems, while the cold-colored beads correspond to the amorphous parts of the chains. The cyclic chains fold back, re-enter the crystal domains, and align with surrounding stems to form the lamellar crystals. A single chain can contribute to multiple lamellae with different principal axis. Most loops are tight folds (hairpins) related to adjacent reentry behavior. By investigating the distribution of amorphous E


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Figure 6. Snapshots of three cyclic chains from sample C2 in the crystallized state at T = 0.6, plotted using the atomistic configuration viewer Atomeye.52 The warm color corresponds to larger bond angle, while the cold color corresponds to smaller bond angle.

Figure 7. Left: radius of gyration (Rg) of individual chains as a function of temperature (T) during cooling/heating cycle. Right: effective persistence length, C*∞ as a function of temperature (T) during cooling/heating cycle. Blue and red line represent the cooling/process, respectively.

or to C∞ * . This implies a reorganization of the chain conformations involving a combination of crystalline and amorphous parts: As temperature is rising, the disordered segments have more mobility to adjust their orientations to align with the stems in crystal regions, accompanied by stretching, and resulting in higher values of Rg. Stem length and C∞ * which are directly correlated with the degree of order in the crystalline phase simply decrease during the annealing process. After melting, Rg drops sharply, and cyclic and linear polymer collapse into random conformations. For all temperatures the values of Rg for cyclic polymers are significantly lower than that of linear polymers with the same chain length. Comparing C2 and L, the volume of gyration of the linear chains of same degree of polymerization, Rg3, is about 4−5 times larger than the volume of cyclic chains. Typical snapshots of two individual chain conformations are displayed in Figure 9a for the C2 system. During crystallization groups consisting of several aligned stems are formed within the same chain (folding motifs), and different parts of the same chain are involved in different crystalline lamellae. Folding motifs of a given chain can be connected by disordered sequences. The secondary structure of a given chain can be

defined by the ordered set of folding motifs and disordered sequences along the contour. The angular distribution for the two visualized chains is displayed in the upper part of Figure 9b as a function of the monomer index during the cooling sweep (time proceeds downward the temperature axis). The starting monomer with number “0” is chosen arbitrarily, and the symmetry mod(1000) is held for the cyclic chain. Bright spots, which correspond to the larger angles, indicate high alignment of neighboring monomers, while dark spots, which represent sharper angles, indicate looplike sequences. Below TC one can observe a band structure which indicates the ordering of stems in folding motifs. This observation is very similar to that for linear chains.40,44 The band structure shows that a chain displays several nonconnected folding motifs. The number of stems per motif is nearly conserved while its width (stem length) grows after first appearance. This is in stark contrast to the “reeling-in” concept13 where the stem length should be conserved while the number of stems in a folding motif should grow linearly. Entanglements. In our previous work we have shown that the entanglement length as calculated by primitive path methods can be directly related with the properties of the F


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folded chain crystal. In particular, the stem length is selected according to the local entanglement length,18 which provides an alternative point of view on the thickness selection problem. As pointed out in the Introduction, the thermodynamic effect (higher supercooling due to decrease in melt entropy), on the one hand, and the geometrical impact of the entanglement network, on the other, give rise to opposite tendencies for the lamellar thickness. We have already shown that cyclic polymers develop thicker lamellae and higher crystallinity under the same thermal history as their linear counterparts, which supports the view of entanglement-controlled structures or “frozen topology”. In the following we report on our analysis of the entanglement structure using the PPA method. The entanglement length is calculated as described in the Model and Methods section. The results for N0e obtained during the cooling−heating cycle are shown in Figure 10. In the melt state we obtain very large values of roughly 1/3 of the contour length of the ring polymers. For linear chains a much higher degree of entanglement is obtained, where the entanglement length, Ne, is about 30 monomers.6,40 This indicates that cyclic polymers are almost nonentangled, as measured by the PPA method using two arbitrary fixed points along the chain’s contour. This result should be reconsidered since the primitive path of the cyclic polymer is squeezed in the melt above the entanglement threshold, i.e., for N ≫ Ne, as compared with that of the linear chain; see for illustration of the concept the lower part of Figure 8. Here, we call Ne the true entanglement

Figure 8. Upper part shows a sketch of a cyclic and a linear polymer in an array of obstacles as a possible way to rationalize entanglement constraints. The lattice-tree model for cyclic chains is adopted from Rubinstein53 and leads to more compact conformations as compared to linear chains. The lower part displays the corresponding lattice graphs of the conformations. Only for the linear chain the lattice graph also corresponds to the primitive path. A primitive path spanning the cyclic molecule, which would be obtained by fixing the monomers (a) and (b), is indicated by the thick dashed line. By applying the PPA, only the dashed path would survive since the remaining loops (red lines) are completely retracted by fixing (a) and (b). If another monomer (c) is fixed, the total primitive path is enlarged as indicated by the thin dashed line. By contrast, the total primitive path of the linear chain would not change by adding additional fixed points.

Figure 9. (a) Snapshots of two cyclic chains A (blue) and B (yellow) from sample C2 during cooling. The translucent beads display their real conformations while the solid tubes represent their primitive paths after applying PPA. The temperatures at which the snapshots have been taken were T = 1.0, 0.9, 0.8, 0.7, and 0.6 (from left to right). (b) Angular values (gray scale) between adjacent bonds for chain A (left panels) and chain B (right panels) during continuous cooling calculated from MD trajectories (upper panels) as a function the monomer index. White belts at lower temperatures correspond to crystallized stems. The lower panels display the same information but for the primitive paths after PPA. The dark lines, contributed by the bonds connected via sharper angles, correspond to kinks in primitive paths. The monomer index is starting with “0” at one of the fixed points of the PPA. The corresponding gray scale maps are shown at the right with angular values given in radians. Angular values smaller than the given range are displayed in black. The red arrows mark the location of typical dark lines (kinks). G


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length, also for cyclic polymers. The simplest way to illustrate this is to consider the lattice-tree (or lattice-animal) model for cyclic polymers as proposed by Rubinstein,53 which is sketched in the upper left part of Figure 8. In this model the cyclic polymer adopts an amoeba-like folded conformation in the presence of obstacles formed by the other chains. This model can be further reduced to a lattice polymer in the dual lattice, as shown in the lower part of Figure 8. On the right-hand side of Figure 8 we have sketched the case of linear chains for comparison. The radius of gyration of the cyclic polymer can be estimated using the concept of hyperbranched polymers55 mapped to the scale of the entanglement constraints. Rg =

⎛ 2 ⎞1/2 2⎜ Nl ⎟ αl p ⎜ 2 ⎟ ⎝ lp ⎠

= αl 2(NNe)1/2 (2)

lNe1/2

where l gives the Kuhn length, lp = denotes the step length of the primitive path, and α is a numerical constant. The numbers N (degree of polymerization) and Ne are given in units of the Kuhn length. This should be compared with the extension of a linear chain in the melt given by Rg2 = l2N/6. The primitive path which is obtained in the lattice-obstacle model is sketched in the lower part of Figure 8. While the lattice path of the linear chain coincides with the primitive path this is not the case for the cyclic chain. Here, the primitive path corresponds to the shortest path across the structure as indicated by the dashed line in the lower left part of Figure 8. The length of the primitive path, Lp, should then by related with the radius of the polymers by Lplp = 6Rg2. Using this concept, we obtain the following relation for the average number of monomers between two kinks in the primitive path:

Ne0 =

1 (NeN )1/2 6α

(3) 2

22

Combining eqs 2 and 3, we obtain Rg = 6α l

N0e

l 2N

Ne0

=

Rg2 R g,L 2

Ne0 (4)

This defines the relation between the apparent entanglement length, N0e , and the true entanglement length as used in the lattice-tree model. The factor between both is the shrinking factor (Rg2/Rg,L2) of the cyclic polymer if compared to the linear counterpart with the same degree of polymerization. For the system C2 we have N0e ≃ 300, Rg2 ≃ 90, and Rg,L ≃ 234, and thus we obtain Ne ≃ 115. This is still about 3 times larger than the entanglement length in the linear system. However, our approach is based on the most simple model for the cyclic melt. Recently, Grosberg has predicting a scaling relation for the primitive path in cyclic polymers using similar arguments.56 It is interesting to note that, in contrast to linear chains, the total primitive path would be extended by adding an additional fixed point such as illustrated in Figure 8 by the point (c). Since in this case the loop ending at (c) cannot be retracted to the original primitive path between (a) and (b), this can explain our findings about the tightening of the entanglement network during crystallization in cyclic polymers as will be discussed further below. Because of the constant prefactor in eq 3, the variation of N0e during the cooling/heating protocol reflects the variation of the true value of Ne as well. Therefore, we will use the original observable N0e to discuss the change of the state of entanglement during the heating/cooling processes in the following. During cooling in the melt state (above the crystallization temperature), N0e decreases for all systems, as can be seen in Figure 10. This can be explained as the consequence of the * which leads to tube thinning. That was increase of stiffness C∞ the key to explain the lamellar thinning for higher supercooling in linear polymer melts using the topological argument.21,40 With the onset of crystallization, the behavior for cyclic polymers deviates from their linear counterpart. The N0e −T curves are monotonous and show a weak further decay during cooling below Tc for cyclic polymers. This is in stark contrast with the sudden increase of Ne for linear chains which is related to a partial disentanglement process. During heating, the behavior of N0e of rings is opposite to that of linear chains. The value of N0e continues to slightly decay until complete melting for cyclic chains. By contrast, reorganization processes in linear chain crystals give rise to further disentanglement; see the lower panel of Figure 10. After melting, N0e begins to increase for cyclic chains, and its value eventually reaches the value before cooling. By contrast, entanglements in linear chain crystals are reestablished after melting which results in a sudden drop of the entanglement length in the fully molten state. To check the role of chain ends during heating, we also cut all the cyclic polymers in the C2 system once in order to form linear polymers immediately after the cooling process. Using this equivalent linear system, we have performed the same heating protocol. Results of these simulations (not shown here) reveal no changes as compared to the cyclic system. This indicates that once the entanglement state is set by crystallization, it is preserved during the complete heating and melting process, regardless of the overall topology of the chains. It is obvious that the values of N0e (and also of Ne according to the calculation above) for cyclic polymers are much higher than that of linear polymers. The snapshots in Figure 9a,b also

Figure 10. Weight-averaged entanglement length (N0e , Ne) versus temperature (T) during cooling/heating cycle. Blue and red lines represent the cooling/heating process, respectively.

2

6R g 2

and thus H


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Macromolecules support this. At T = 1.0, the primitive paths of two chains show few kinks only, although their conformations (before PPA has been applied) are randomly coiled. However, it is interesting to note that the number of kinks in the primitive paths is increasing for lower temperatures. In the lower panel of Figure 9b, we monitor the crystallization process during continuous cooling by comparing the angular values between successive bonds in the chains of MD trajectories and those in PPA conformations. Moving down the vertical axis (to lower temperatures) corresponds to the time elapsed during cooling. The black lines in the lower panels represent the PPA kinks. Two vertical black lines at monomer index equal to 0 and 500 correspond to the fixed monomers (pseudoends). We note that changing the location of these pesudoends within the chain has little influence on the number and locations of kinks, which thus appear robust with respect to the choice of pesudoends. The PPA kinks persist during continuous cooling. Let us now point to a fact that might have essential consequences for the state of entanglement during crystallization and annealing of cyclic polymers: Because of crystallization monomers are fixed in crystalline lamellae and thus provide fixed points of the loops in the amorphous phase. Apparently, this can lead to the formation of additional entanglement constraints as can be clearly observed in Figure 9b for chain B. Here, a wide deep-dark region emerges as soon as the crystallization occurs. In both panels one can observe the increase of the number of dark lines (which corresponds to kinks in the primitive path) as crystallization proceeds. In a number of supplementary simulations (not shown here), we have studied the effect of fixing additional monomers in the rings during PPA in the melt state. This mimics the fixation of monomers in the crystal phase during cooling. We always found a strong increase of the number of entanglements after subtracting the additional kinks caused by the fixed points. This offers an interesting explanation for the very different behaviors of the entanglement lengths for cyclic and linear chains during crystallization and annealing as obtained in Figure 10: the formation of additional topological constraints due to the emergence of the crystalline phase which grafts the end points of the amorphous sequences between two entry points of the crystalline domain. This leads to a prolongation of the primitive path in cyclic polymers as sketched in the lower left part of Figure 8. This, in turn, decreases the effective entanglement length (ratio of the chain length and the length of the primitive path). Cyclic polymers were thus nucleated under the condition of large entanglement lengths, and their thickness exceeds the value which would have been obtained under conditions of stronger entanglement. This mechanism of tightening the entanglement network during crystallization is most likely responsible for the missing reorganization (thickening) of the lamellae during heating and subsequently suppresses further disentanglement, too. Linear polymers, by contrast, form their lamellae under the condition of high entanglement density which is not increased by fixing of monomers in the crystalline domains; see lower left side of Figure 8. The fixation of monomers of entangled chains that simultaneously crystallize at the same or at different crystalline domains is an essential argument to understand the role of entanglement in thickness selection. Even very slow crystallization processes would not be able to disentangle the material unless the crystallization rate is so small that in average not a second monomer of the same chain crystallizes before a

retraction/reptation cycle of the chain is completed. For one monomer fixed in the crystal this can even imply an exponentially long time scale, τd ∼ exp{N/Ne}. Though the cooling and growth rates in simulations are much larger than in experiments, both should belong to the same class in this respect. Assuming for the moment that polymer crystallization from the melt state always occurs on time scales which preserved (fixed) entanglement constraints, the relevant time scale is that of the reorganization of the entanglement network by changing temperature and not the reptation time (which loses its meaning here after the fixation of more than one monomer in the chain). As shown in our previous work57 indeed very rapid changes, even in terms of simulations, are necessary to preserve the state of entanglement upon change of temperature. The thicker lamellae formed by cyclic polymers is correlated with the higher melting points as we reported in the Thermodynamic Properties section. We note the similarity of our results concerning the correlation between the lower entanglement density and the increase of melting temperatures to the experimental work for linear partially disentangled UHMWPE whose large molar mass reduces the effects chain ends substantially.17 Two separate endothermic melting peaks have been reported there, where the higher one can be associated with the less entangled domains, while the lower one to the entangled domains.17

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CONCLUSIONS Although long cyclic polymers differ from their linear counterparts only by the absence of chain ends, we found a much enhanced crystallization behavior. The crystallization and melting points as well as the crystallinity are substantially higher for cyclic chains as compared to linear chains. In addition, the stem lengths are larger as compared to their linear counterparts under the same thermal history, and the crystallinity is increased up to 100% in the cyclic systems. The difference between cyclic and linear polymers should be attributed to direct impact of the entanglement state on the crystalline properties (since kinetic models based on the effective supercooling would lead to the opposite conclusion). The latter can be quantified in computer simulations by primitive path analysis (PPA) which has been performed at all stages during cooling, crystallization, annealing, and melting in the present work. Notably, PPA predicts a nearly untangled state of cyclic polymers in the melt. However, during crystallization creation of entanglement constraints due to monomers being absorbed and trapped in crystal lamellae can be observed. This is related with a decrease of the entanglement length/increase of the length of the primitive path during crystallization and heating and is in marked contrast to linear chains where a disentanglement process dominates during crystallization and annealing. Related to this observation we found that cyclic polymers do not show an increase of the stem length during heating, again in contrast with the observation for linear systems which display pronounced reorganization behavior in this stage. On the other hand, the application of the PPA for cyclic polymers needs reconsideration. Although there is no dynamical plateau observed for purified cyclic nonconcatenated polymer melts, it is usually assumed that also cyclic polymers are entangled.9 This idea is not misleading if one considers smaller sequences within the cyclic chains. Pinning their ends would give rise to local entanglement constraints quite I


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Macromolecules

(2) Cates, M.; Deutsch, J. Conjectures on the statistics of ring polymers. J. Phys. (Paris) 1986, 47, 2121−2128. (3) Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. I. Statics. J. Chem. Phys. 2011, 134, 204904. (4) Lang, M.; Fischer, J.; Sommer, J.-U. Effect of Topology on the Conformations of Ring Polymers. Macromolecules 2012, 45, 7642. (5) Obukhov, S. P.; Rubinstein, M.; Duke, T. Dynamics of a ring polymer in a gel. Phys. Rev. Lett. 1994, 73, 1263. (6) Halverson, J. D.; Lee, W. B.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Molecular dynamics simulation study of nonconcatenated ring polymers in a melt. II. Dynamics. J. Chem. Phys. 2011, 134, 204905. (7) Halverson, J. D.; Grest, G. S.; Grosberg, A. Y.; Kremer, K. Rheology of ring polymer melts: from linear contaminants to ringlinear blends. Phys. Rev. Lett. 2012, 108, 038301. (8) Everaers, R.; Sukumaran, S. K.; Grest, G. S.; Svaneborg, C.; Sivasubramanian, A.; Kremer, K. Rheology and microscopic topology of entangled polymeric liquids. Science 2004, 303, 823−826. (9) Kapnistos, M.; Lang, M.; Vlassopoulos, D.; Pyckhout-Hintzen, W.; Richter, D.; Cho, D.; Chang, T.; Rubinstein, M. Unexpected power-law stress relaxation of entangled ring polymers. Nat. Mater. 2008, 7, 997. (10) Fischer, E. W. Stufenförmiges und Spiralförmiges Wachstum bei Hochpolymeren. Z. Naturforsch., A: Phys. Sci. 1957, 12, 753−754. (11) Keller, A. A Note on single crystals in polymers - Evidence for a folded chain configuration. Philos. Mag. 1957, 2, 1171. (12) Su, H.-H.; Chen, H.-L.; Díaz, A.; Casas, M. T.; Puiggalí, J.; Hoskins, J. N.; Grayson, S. M.; Pérez, R. A.; Müller, A. J. New insights on the crystallization and melting of cyclic PCL chains on the basis of a modified Thomson-Gibbs equation. Polymer 2013, 54, 846−859. (13) Hoffman, J. D.; Lauritzen, J. I.; Passaglia, E.; Ross, G. S.; Frolen, L. J.; Weeks, J. J. Kinetics of Crystallization from Solution and the Melt. Kolloid Z. Z. Polym. 1969, 231, 564−592. (14) Sadler, D. M. New explanation for chain folding in polymers. Nature 1987, 326, 174−177. (15) Armistead, K.; Goldbeck-Wood, G. Polymer Crystallization Theories. Adv. Polym. Sci. 1992, 100, 219−312. (16) Rastogi, S.; Lippits, D. R.; Peters, G. W.; Graf, R.; Yao, Y.; Spiess, H. W. Heterogeneity in polymer melts from melting of polymer crystals. Nat. Mater. 2005, 4, 635−641. (17) Liu, K.; de Boer, E.; Yao, Y.; Romano, D.; Ronca, S.; Rastogi, S. Heterogeneous Distribution of Entanglements in a Nonequilibrium Polymer Melt of UHMWPE: Influence on Crystallization without and with Graphene Oxide. Macromolecules 2016, 49, 7497. (18) Luo, C.-F.; Sommer, J.-U. Frozen topology: Entanglements control nucleation and crystallization in polymers. Phys. Rev. Lett. 2014, 112, 195702. (19) Luo, C.-F.; Kröger, M.; Sommer, J.-U. Entanglements and crystallization of concentrated polymer solutions: molecular dynamics simulations. Macromolecules 2016, 49, 9017. (20) Luo, C.-F.; Kröger, M.; Sommer, J.-U. Molecular Dynamics Simulations of Crystallization under Confinement: Entanglement Effect. Polymer 2017, 109, 71. (21) Luo, C.-F.; Sommer, J.-U. The role of thermal history and entanglement related thickness selection in polymer crystallization. ACS Macro Lett. 2016, 5, 30. (22) Pérez-Camargo, R. A.; Mugica, A.; Zubitur, M.; Müller, A. J. Crystallization of cyclic polymers. Adv. Polym. Sci. 2015, 276, 93−132. (23) Yu, G. E.; Sun, T.; Yan, Z. G.; Price, C.; Booth, C.; Cook, J.; Ryan, A. J.; Viras, K. Low-molar-mass cyclic poly(oxyethylene)s studied by Raman spectroscopy, X-ray scattering and differential scanning calorimetry. Polymer 1997, 38, 35−42. (24) Tezuka, Y.; Ohtsuka, T.; Adachi, K.; Komiya, R.; Ohno, N.; Okui, N. A defect-free ring polymer: size-controlled cyclic poly(tetrahydrofuran) consisting exclusively of the monomer unit. Macromol. Rapid Commun. 2008, 29, 1237−1241. (25) Córdova, M. E.; Lorenzo, A. T.; Müller, A. J.; Hoskins, J. N.; Grayson, S. M. A comparative study on the crystallization behavior of

indistinguishable from that of linear chains. Also, it appears counterintuitive to refuse the packing constraint argument in this case which is based on local conformational properties. On the basis of a simple lattice-tree model for cyclic polymers, we have derived a relation between the true entanglement length and the direct result from the primitive path analysis. This model can also explain the increase of the apparent entanglement density by fixing monomers such as it occurs during crystallization. The estimated entanglement length is still much larger than the corresponding result for linear chains. Our simulations point to the dominating role of the state of entanglement for polymer crystallization including thickness selection. Already in previous studies of linear melts we have shown by using in situ correlation analysis that the local state of entanglement correlates with the local stem length.18 This has been confirmed in the analysis of melts diluted with very small chains.20 For linear chains a characteristic folding motif with a nearly constant number of folds has been detected. Compared to these results, a higher degree of folding per entanglement length seems likely for cyclic polymers. Here, NMR studies similar to the those carried out by Miyoshi et al. for linear polymers could help to clarify the picture.58 On the other hand, we find an enhanced fraction of short loops as compared with the linear system indicating a larger number tight hairpin-like folds. Analysis of chain folding in the crystalline state reveals that individual cyclic chains participate in different crystalline domains partially with random orientations, too. This resembles a frozen-in lattice-tree structure as proposed for the conformations of cyclic polymers in the melt. To conclude, our results show clearly that strictly nonconcatenated cyclic polymers increase the order in the crystalline state as compared to their linear counterparts under the same thermal history. Our results also give indications that the crystalline morphology is controlled by the entanglement constraints in long cyclic polymer with the particular feature to tighten the constraints during crystallization. This will be further clarified in future work where mixed systems composed of the linear and cyclic chains are considered.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (C.L.). *E-mail: [email protected] (D.Y.). *E-mail: [email protected] (J.-U.S.). ORCID

Chuanfu Luo: 0000-0002-4911-7870 Jens-Uwe Sommer: 0000-0001-8239-3570 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge financial support by the National Natural Science Foundation of China (NSFC) 21374011 and 21434001 and computing time from ZIH at TU Dresden.

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REFERENCES

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Molecular Dynamics Simulation of Crystallization Cyclic Polymer Melts

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