Monte Carlo Simulations of Strong Memory Effect of Crystallization in

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Monte Carlo Simulations of Strong Memory Effect of Crystallization in Random Copolymers Huanhuan Gao,† Madhavi Vadlamudi,‡,§ Rufina G. Alamo,‡ and Wenbing Hu*,† †

Department of Polymer Science and Engineering, State Key Laboratory of Coordination Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing, China 210093 ‡ Department of Chemical and Biomedical Engineering, FAMU/FSU College of Engineering, Tallahassee, Florida 32310-6046, United States S Supporting Information *

ABSTRACT: Recently, experiments reported a strong memory effect of crystallization in model ethylene-based homogeneous random copolymers after being annealed at temperatures higher than the equilibrium melting point of copolymers. By means of dynamic Monte Carlo simulations of random copolymers, we reproduced this phenomenon in the similar model copolymer systems. We attributed this phenomenon to the sequence-length segregation upon first-time crystallization. The resulting heterogeneous melt of copolymers survives upon annealing below the critical demixing point that could be much higher than the equilibrium melting point of copolymers. Therefore, the local high concentration of long sequences raises the local melting point to accelerate primary crystal nucleation upon second-time crystallization. This source of memory effects demonstrates how crystallization can be influenced by the substantial trend of demixing between different sequences in homogeneous random copolymers.

I. INTRODUCTION Crystallization behaviors of polymers depend very much upon their thermal history and often exhibit a memory to the previous crystallization process.1 For instance, a thermal treatment of homopolymers after first-time crystallization may leave small traces of survived crystallites, oriented segments, or less-entangled polymer chains,2−6 resulting in higher crystallization rates, higher crystallization temperatures, smaller spherulites, or polymorphic modifications upon second-time crystallization.7−10 Therefore, the memory effect is an important issue in the study of polymer crystallization, which facilitates our better understanding on controlling the morphology and property of polymer crystals in the industrial processes.11,12 Industrial polymers are often multicomponent systems containing statistically random defects along the chain due to polymerization of variable chemical species, asymmetrical geometries, or stereoisomerisms, making their variable crystallization behaviors with versatile physical properties for a wide spectrum of practical applications. Good examples are the ethylene-based statistical copolymers, in particular, linear low density polyethylene.13,14 Statistical copolymers can be classified into heterogeneous copolymers and homogeneous copolymers, according to their distribution characters of comonomers among macromolecules.15 Heterogeneous copolymers are usually obtained by the addition copolymerization with multisite Ziegler−Natta catalysts or by a batch copolymerization with drifting feed compositions which harvests variable comonomer contents between the early and later stages of polymerization. Homogeneous copolymers are © XXXX American Chemical Society

prepared by a continuous copolymerization process with constant feed compositions and a single-site catalyst. The latter also includes hydrogenated polybutadienes because of the same random distribution characters of chemical defects on the microstructures.16,17 In our previous dynamic Monte Carlo simulations of copolymer crystallization, heterogeneous copolymers exhibit an obvious monomer/comonomer demixing prior to first-time crystallization.18,19 Heterogeneous copolymers with major components containing two extreme compositions behave like a polymer blend. The prior demixing between two components can be attributed to their different crystallizability, which is a result of interplay between polymer crystallization and liquid−liquid demixing.20−25 Hu and co-workers have derived the mixing free energy of polymer blends in such a case, as given by26 Δfmix k bT

=

ϕ1 N1

ln ϕ1 +

⎡ B ln ϕ2 + ϕ1ϕ2⎢(z − 2) ⎢⎣ k bT N2 ϕ2

2 ⎤ ⎛ 2 ⎞⎛ 1 ⎞ Ep ⎥ + ⎜ 1 − ⎟ ⎜1 − ⎟ ⎝ z ⎠⎝ N1 ⎠ k bT ⎥⎦

(1)

where ϕ and N are the volume fractions and chain lengths for polymer component 1 and 2, respectively, z is the coordination Received: April 24, 2013 Revised: July 24, 2013

A

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number in the lattice space, B is the site−site mixing interaction between two components, Ep is the energy parameter for parallel packing of polymer bonds, which belongs only to the component 1 to characterize the different crystallizability of two components, kb is the Boltzmann constant, and T is the temperature. One can see from eq 1 that longer chains as well as symmetric compositions intend to contribute minimum mixing entropy that favors demixing. In addition, even with B = 0, the interaction parameter Ep alone has already made the critical demixing point of binary blends much higher than the equilibrium melting point of homopolymers.22 In heterogeneous random copolymers, due to strong demixing prior to crystallization, the memory effect of crystallization appears similar to homopolymers and ends at the annealing temperatures below the equilibrium melting points of homopolymers.27 Crystallization of homogeneous copolymers appears more sensitive to the sequence distributions than that of heterogeneous copolymers.28 With the increase of comonomer contents, the amount of long monomer sequences decreases exponentially, leading to thinner lamellar crystals as well as lower melting points, with a rapid decay of crystallinity and lamellar morphology.16,29 Early in the 1950s, Flory has derived a rigorous theoretical relationship between the equilibrium melting point (Te) and the comonomer content (the mole fraction f 2), based on the assumption of a pure crystal phase of crystallizable monomers, as given by 1/Te − 1/Te0 = −(R /ΔHu) ln(1 − f2 )

chemical natures. In principle, those long sequences hold a substantial trend of demixing from those short sequences, like in the extreme cases of heterogeneous copolymers or binary blends. However, because of a strong topological restriction along chain connection and a well distribution of sequence lengths, homogeneous copolymers perform neither macroscale phase separation nor nanoscale microphase separation. In 1992, de Gennes proposed weak segregation of homogeneous statistical copolymers under a large Flory−Huggins parameter χ.34 Molecular simulations have demonstrated that there exists a significant selection of sequence lengths upon crystallization of homogeneous random copolymers.35 The sequence-length selection during first-time crystallization will enhance weak segregation, resulting in a heterogeneous melt of copolymers that can only be eliminated above the critical point of demixing (could be very much higher than the equilibrium melting point of copolymers and even of homopolymers). Thus, the strong memory effect can be attributed to the residual sequence-length segregation yielded by first-time crystallization, which restricts the diffusion of sequences toward the homogeneous melt of copolymers. In other words, after the melting of large crystallites, the long sequences remain accumulated in the local area, which raise the local melting point above the global equilibrium melting points averaged over all the sequence lengths. Higher melting points make an enhancement of primary nucleation and even allow the survival of self-seeding nuclei provided under not high enough annealing temperatures, resulting in a strong memory effect of crystallization in homogeneous random copolymers. In this paper, by means of dynamic Monte Carlo simulations of homogeneous random copolymers, we reproduced the strong memory effect above with the parallel procedures. We first obtained the equilibrium melting points of homogeneous random copolymers and then compared them to the uppercritical annealing temperatures to reproduce the memory effect above the equilibrium melting point of copolymers. After that, we made a careful structural analysis on the samples right before second-time crystallization to evidence the segregation state of long sequences upon annealing at high temperatures, which could be responsible for the observed strong memory effect in second-time crystallization.

(2)

Here, T0e is the equilibrium melting point of homopolymers, R is the gas constant, and ΔHu is the heat of fusion per mole of chain units.30 Recently, Alamo’s group performed DSC scanning experiments to study the memory effect of ethylene−1-butene homogeneous random copolymers.31 The copolymer samples were cooled with a constant cooling rate for first-time crystallization from the completely molten state, and then the samples were annealed isothermally at a certain high temperature for a long enough time, followed with secondtime crystallization in the second cooling with the same cooling rates. It has been found that the shifting-up of the crystallization peak in the second cooling remains at the annealing temperatures above the Flory’s equilibrium melting point of copolymers (and even above the equilibrium melting point of homopolymers), subject to the sample molar mass higher than 4500 g/mol and both the comonomer content and the end temperature of the first cooling not too high.31 Such a strong memory effect will not be erased upon a significant elongation of annealing time at the temperatures above the equilibrium melting point of copolymers.31 This result implies that the effect cannot be attributed to any dynamic reasons such as remaining orientations in the melt. To the best of our knowledge, the only observation on the strong memory effect above the equilibrium meting point has been reported in nylon.32,33 In that case, the strong hydrogen-bonding structures may remain in the melt. The strong memory effect of homogeneous random copolymers cannot hold the same reason as nylon and thus be worthy of further investigation. The memory effects of copolymer crystallization could be raised by the demixing of different components induced by first-time crystallization. For homogeneous copolymers, the different components in eq 1 refer to the monomer/ comonomer sequences with their different crystallizability and

II. SIMULATION TECHNIQUES A. Sample Preparation. In our simulations, each polymer chain consecutively occupies 128 lattice sites, and each occupied lattice site represents either a monomer or a comonomer according to an ideal addition copolymerization process for making a series of homogeneous random copolymers with preset comonomer fractions f 2 varied from 0.06, 0.12, 0.24, 0.36 to 0.44.18 We put 1920 polymer chains into a 643 cubic lattice box with periodic boundary conditions. The occupation density of the lattice box was as high as 0.9375 to mimic the bulk polymer phase, and the minor vacancies represented the free volume for the microrelaxation of polymer chains in our simulations. B. Microrelaxation Model and Metropolis Sampling Algorithm. Polymer chains in the lattice space were performing single-site jumping for their microrelaxation of chain dynamics.36 Each randomly chosen site on the chain jumped into a randomly chosen neighboring vacancy site, sometimes with partial sliding diffusion along the chain.37 During the single-site jumping, the hard-core volume exclusion of both polymer beads and bonds was considered. The mixing B

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process, the initiation of crystallization from a homogeneous bulk phase requires for primary nucleation, and the latter is mainly responsible for a significant supercooling on cooling. We introduced a crystalline layer as a template of crystal nuclei to exempt the primary nucleation process and thus to avoid the significant supercooling on cooling. In principle, the onset crystallization temperature on cooling in the sample system with a template should be in the vicinity of the equilibrium melting point. We put a fixed template layer formed by eight regularly folded homopolymer chains with 16 monomers of the stem length in the copolymer melt, as demonstrated by the snapshot in Figure 1a. After a long-term athermal relaxation to obtain the

interactions between monomers and comonomers were set as zero, and the crystallization interactions between monomer bonds were considered in the conventional Metropolis sampling algorithm. In the ethylene-based copolymer system, there is no strong interaction between monomer and comonomer units simply because they share very similar chemical compositions. Thus, the only driving force for demixing is the different crystallizability between monomers and comonomers, similar to our previous simulations.18−20 In the Metropolis algorithm, each trial move for single-site jump was accepted by a probability of min[1, exp(−EpΔn/kb/ T)], where Ep is the parallel packing energy of monomer bonds driving polymer crystallization and Δn is the net change in the number of parallel pairs of packing bonds in each step of microrelaxation.38 By a dimensionless treatment, we simply employed the reduced temperature, kbT/Ep, denoted as T in the rest part of this paper. Note that the parallel packing energy Ep was assigned only to the purely monomer bonds, and in this way the monomers were distinguished from the comonomers in our simulations. Since in ethylene-based copolymers any side groups of comonomers are larger than the hydrogen atoms of ethylene sequences, which make comonomers excluded from the crystalline lattice of ethylene sequences, we have to implement this protocol in our simulations: a sliding diffusion of comonomer units along the chain into the parallel-packing pair of monomer bonds is forbidden. C. Simulation of DSC Scanning Experiment. In the lattice space, the maximum number of neighboring parallel packing pairs around one monomer bond is 24 (counting the neighbors along both lattice axes and (face and body) diagonals, except for two neighbors along the chain). We arbitrarily set a criterion of parallel bonds at five to distinguish the crystalline bond from the initially amorphous bond. In other words, if one monomer bond contained more than five parallel neighboring monomer bonds, it was recorded as a crystalline bond; otherwise, it belonged to an amorphous bond. Then, we defined the crystallinity (noted as Φ) of the sample system as the fraction of crystalline bonds in all the monomer bonds. It should be noted that this definition of crystallinity was different from the relative crystallinity measured in experiments where the fraction was in the total amount of monomers rather than in the monomer bonds. Mimicking the temperature-scanning programs of DSC experiments, we stepwise changed the temperature with the step length 0.01. During each step, we carried out 300 Monte Carlo cycles, where one Monte Carlo cycle (MCc) was defined as the total amount of trial moves equal to the amount of occupied sites in the lattice. The first 100 MCcs at each temperature were discarded for system relaxation. The reported crystallinity data were averaged over the subsequent 200 MCcs. By monitoring the time evolution of crystallinity during a cooling process, we could calculate the crystallization rate from the first derivative of the crystallinity curve. The crystallization rate in our simulations corresponds to the heat flow rate contributed purely by the release of latent heat upon thermal fluctuations or crystallization in DSC experiments.

Figure 1. (a) Snapshot of the initial bulk copolymers with a crystallization template formed by folded long chains. Polymer bonds are drawn as tiny cylinders. The template is painted yellow, while the amorphous copolymer chains are painted blue. (b) Temperatureevolution curve of crystallization rates for a random copolymer with comonomer content 0.06 and chain length 128, on cooling with the rate 0.01 per 300 MCcs. The tentative equilibrium melting temperature was recorded by the crossover of two extrapolated dashed lines.

amorphous copolymer sample, the temperatures were stepwise dropped down from 6 to 1 by the step length 0.01 per 300 MCcs. The time evolution of crystallization rates was recorded to find the onset temperature where it deviated from the horizontal baseline, as demonstrated for example in Figure 1b. This temperature was regarded tentatively as the equilibrium melting point of homogeneous copolymers similar to the Flory’s theoretical approach because crystallization was initiated from the template toward the nearby homogeneously distributed monomer sequences, and crystal growth is fast

III. SIMULATION RESULTS A. Equilibrium Melting Temperatures. A direct way to measure the equilibrium melting points (Te) could be the computation of free energy change, which however appears formidable for such a large system in our current simulations. Therefore, we employed an indirect way.34 During a cooling C

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enough to avoid any long-distance diffusion due to the substantial trend of demixing. The equilibrium melting points Te for copolymers with various comonomer contents are summarized in Figure 2 for comparisons to the upper critical annealing temperatures Tc of the memory effect obtained by the below-described procedure.

Figure 4. Temperature evolution of crystallization rates of a copolymer with f 2 = 0.44 and chain length 128 during the secondtime crystallization with various annealing temperatures (Ta) as denoted. Copolymers have annealed at a given temperature for 10 000 MCcs after the first-time crystallization. The dashed vertical line indicates the fixed high temperature for comparison of crystallization rates. Figure 2. Comparison between the upper-critical annealing temperatures of the memory effect (Tc) and the equilibrium melting temperatures (Te) of homogeneous copolymers with various comonomer contents.

beginning of second-time crystallization, for instance, T = 2.4, crystallization rates are shifted up by a memory effect if the annealing temperatures are not high enough. The acceleration disappears with further increase of annealing temperatures. Therefore, we compared the crystallization rates at the fixed high temperature T = 2.4 for the samples with various annealing temperatures to obtain the upper-critical annealing temperature of such a memory effect, as demonstrated in Figure 4. Of course, the selection of the fixed high temperature depends on the copolymer with variable comonomer contents. The results for copolymers with various comonomer contents are summarized in Figure 5a. The upper-critical annealing temperature (Tc) can be recorded at the crossover of two extrapolated lines. We have examined a slight shift in the selection of the fixed high temperature for the comparison of the crystallization rates, and it is proved to have little influence on the result of the upper-critical annealing temperatures. We performed the same procedure above for the copolymers with various comonomer contents and extended the annealing time from 10 000 to 50 000 MCcs. The upper-critical temperatures are basically the same, as shown in Figure 5b. Thus, the memory effect can be attributed to a thermodynamic property of homogeneous random copolymers rather than a dynamic effect caused by a lack of long-enough annealing. In order to make sure that the acceleration of second-time crystallization comes from the memory effect of first-time crystallization, we added a high-temperature melting process right after first-time crystallization to erase its trace completely, while the rest part of the temperature-scanning program was kept as the same as above, as demonstrated in Figure 6a. In this situation, the crystallization rates during the second-time crystallization are shown in Figure 6b. There is almost no difference among the samples at the fixed high temperature T = 2.4 after being annealed at various high temperatures. Therefore, the memory effect of crystallization can be assigned only to the first-time crystallization, rather than some other effects during annealing. In experiments, the strong memory effects disappear when the end temperature of the first cooling becomes high enough.31 We correspondingly shifted up the end temperatures

B. Upper-Critical Annealing Temperatures for the Memory Effect. Following the protocol of experiments reported in ref 31, we first cooled the initially amorphous copolymers from temperature six to one for first-time crystallization. Then, we annealed the sample at a given high temperature for 10 000 MCcs, followed with the second cooling process for second-time crystallization. A typical temperature program is summarized in Figure 3. A stepwise

Figure 3. Illustration of the temperature program in mimicking the DSC scanning experiment to observe the memory effect of crystallization in copolymer samples.

jumping to a certain high temperature right after the annealing process was taken in order to save the CPU time on the simulation of the second cooling. Since the temperature was linearly decreased with time during second-time crystallization, the values of −dΦ/dT characterize the crystallization rates. The crystallization rates during second-time crystallization of a random copolymer with comonomer content 0.44 after annealed at different annealing temperatures (T a ) are summarized in Figure 4. One can clearly see that at the D

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Figure 5. Crystallization rates of the copolymers with chain length 128 and various denoted comonomer contents at a selected high temperature during second-time crystallization: (a) annealing time = 10 000 MCcs; (b) annealing time = 50 000 MCcs. The high temperatures for comparison of crystallization rates were selected at T = 3.55 for f 2 = 0.06, T = 3.35 for f 2 = 0.12, T = 2.95 for f 2 = 0.24, T = 2.60 for f 2 = 0.36, and T = 2.40 for f 2 = 0.44. The dotted lines demonstrate the extrapolation to make the crossover as the uppercritical annealing temperatures (Tc), which were marked with the arrows.

Figure 6. (a) Temperature program with a high-temperature melting inserted to erase the trace of the first-time crystallization. (b) Crystallization rates at the fixed high temperature during the secondtime crystallization of a copolymer with f 2 = 0.44 and chain length 128. Different curves present the copolymers with different annealing temperatures as labeled. The dashed line indicates the fixed high temperature.

of the first cooling, and indeed the upper critical temperatures of the memory effects move down toward the equilibrium melting point of copolymers, as demonstrated in Figure 7. One may expect that when the end temperature of the first cooling becomes high enough, crystallization could not make a clear effect of sequence demixing, so the memory effect could no more exist at high annealing temperatures. In Figure 2, for low-comonomer-content samples, no memory effect can be observed above equilibrium melting temperatures, consistent with homopolymers as the extreme case of the lower end of comonomer contents; however, for high-comonomer-content samples, Tc becomes higher than Te, consistent with the experimental results.31 In this sense, our simulations have reproduced the experimental observations on the strong memory effect of crystallization, unique to homogeneous random copolymers. C. Structural Analysis. One of significant advantages of molecular simulations is that all the molecular details have been recorded for further structural analysis to understand the strong memory effect of crystallization in homogeneous random copolymers. We took a copolymer with comonomer content 0.44 as a typical example because of its most significant memory effect in the current study.

Figure 7. Crystallization rates of the copolymers with chain length 128 and comonomer contents 0.44 at various end temperatures of the first cooling as labeled in the figure, obtained at T = 2.4 during second-time crystallization after annealed at various annealing temperatures for 10 000 MCcs. The dashed lines are drawn to guide the eyes.

We first focused on the survival of crystallites on annealing at high temperatures around the upper-critical annealing temperature. We traced the time evolution of one crystal during annealing at Ta = 2.6 (under this annealing temperature, the E

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harvested by first-time crystallization, resulting in a less extent of long-sequence segregation and hence a less shifting-up of local melting points. So no more memory effect will occur at high enough end temperatures of first cooling if annealed at a fixed high temperature, as observed in both experiments and simulations. In order to evidence the scenario above, we traced the structure factors and the demixing parameters in our simulations to check the possible segregation remaining right after the annealing process. The structure factor was defined as the spherically averaged one-dimensional results for monomers, as given by 1 S(q) = 3 ∑ exp(iq·rjk)⟨σ(rj)σ(rk)⟩ 3L j , k (3)

copolymer starts to show the memory effect although above the equilibrium melting point). Copolymers form a typically lamellar crystals right after first-time crystallization (Figure 8a). When the temperature rises up to 2.6 on annealing, the

where q is the wave vector with discrete integers times 2π/L and L is the linear size of the cubic box. The summation took over all pairs of the lattice sites with a distance r. If a lattice pair was occupied by the monomers, σ = 1; otherwise, σ = 0. The demixing parameter was defined as the mean fraction of neighboring sites of each monomer occupied by other monomers, which appeared sensitive to any segregation behavior of monomers. The structure factors under various situations for the copolymers with comonomer content 0.44 are shown in Figure 9. By comparison, one can see that if the annealing temperature

Figure 8. Snapshots of one lamellar crystal of a copolymer with f 2 = 0.44 and chain length 128 during the annealing at Ta = 2.6 above the equilibrium melting point. Crystalline bonds are drawn in yellow cylinders. (a) The initial state before annealing. (b) Annealing time at 2000 MCcs. (c) Annealing time at 5000 MCcs. (d) Annealing time at 10 000 MCcs.

lamellar crystal immediately melts with a significant shrinkage of its shape (Figure 8b). However, it did not melt completely; instead, the crystal size decreases to a small value and becomes stable there with slight fluctuations (Figure 8c,d). We also provided a movie demonstrating this annealing process as Supporting Information. From the snapshots in Figure 8, one can clearly see that small crystalline domains remain after the annealing process. It is well-known that the thermal stability of a crystallite is weakened when its size becomes smaller. Therefore, it is a challenge question why these small crystallites can survive even though the previously larger ones have melted above the equilibrium melting point of copolymers. A logical scenario for the observation above is that the melting of the previously larger crystallite has changed the local environment for the small survived crystallites. We know that during first-time crystallization the crystallite will collect long monomer sequences. The melting of large crystallites will release the long monomer sequences and thus raises their local concentrations as well as the melting point. Long monomer sequences will have a stronger intention to make phase separation in comparison to short sequences for their less mixing entropy according to eq 1 (heterogeneous copolymers can be an extreme case of this trend). The local high concentration of long sequences may be maintained with a potential driving force for phase separation between monomer sequences of extremely different lengths. In other words, firsttime crystallization helps to realize the potential phase separation between monomer sequences of two extremely different lengths; otherwise, they maintain their weak segregation state in the almost homogeneous melt. At higher end temperatures of the first cooling, less crystallinity will be

Figure 9. Structure factors of the copolymer with comonomer content 0.44 and chain length 128, during or after the first-time crystallization with subsequent annealing at the labeled temperatures for 10 000 MCcs.

is not high enough (at 2.6 with a memory effect), a significant peak occurs in the low-q region in comparison to the systems without the memory effect. No such peak occurs at the same temperatures before crystallization during the first cooling process. Therefore, this peak corresponds to the structures survived during annealing after first-time crystallization. In the first cooling, the demixing parameters increase with crystallinity, indicating a segregation induced by first-time crystallization. However, after first-time crystallization, the demixing parameters decay back toward the initial amorphous melt upon annealing at T = 2.6, implying quite weak segregation at the monomer/comonomer-unit level, as shown in Figures 10a,b. Note that the demixing parameter reflects only very local concentration change of monomers rather than a larger scale of concentration change. F

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Figure 10. (a) Temperature evolution of demixing parameters and crystallinity on the first cooling process to monitor the first-time crystallization. (b) Time evolution of demixing parameters on annealing various labeled temperatures for the copolymer with comonomer content 0.44 and chain length 128. The dashed line indicates the demixing parameter level before the first-time crystallization.

Figure 11. Structural analysis of the long-sequence component (six or more monomers) for the copolymers with f 2 = 0.44 and chain length 128. (a) Structure factors of various conditions as labeled during or after first-time crystallization and subsequent annealing at various labeled annealing temperatures. (b) Time evolution of demixing parameters of samples during annealing at various annealing temperatures. The short-dashed line indicates the demixing parameter value before first-time crystallization.

We now focused our attention on the potential segregation at the long-sequence level rather than at the monomer-unit level. To this end, we arbitrarily separated all the monomer sequences into two components according to their sequence lengths: the long-sequence component containing six or more monomers; the short-sequence component containing fewer than six monomers. We redefined the structure factors and the demixing parameters on the basis of the long-sequence component. The results in parallel to Figure 10 above are shown in Figure 11. As shown in Figure 11a, the structure factors of the longsequence component show again the significant peak, at the annealing temperature 2.6 with a strong memory effect. We also separately calculated the structure factor attributed to the crystalline monomers. The structure factors of crystalline monomers are very weak. Therefore, this significant peak should be assigned only to the domain rich with long monomer sequences rather than to the survived crystalline domains. Moreover, the demixing parameters of the long-sequence component show a significant remaining segregation between long and short sequences. We also showed two snapshots of sectional surfaces of the copolymer systems to observe different distributions of long sequences. Figure 12a was taken at T = 2.4 during the first cooling, while Figure 12b was taken again at T = 2.4 during the second cooling right after annealing at T = 2.6 with a strong memory effect. One can clearly see the homogeneous distribution of long sequences in Figure 12a, but the heterogeneous distribution in Figure 12b, as indicated by several small clusters of long sequences. The crystalline

Figure 12. Snapshots of the copolymer samples with f 2 = 0.44 and chain length 128 to show the yellow long sequences (six monomers or longer) at T = 2.4 (a) during first-time crystallization and (b) during second-time crystallization after annealed at T = 2.6 for 10 000 MCcs. The bonds of short sequences (less than six monomers) are drawn as the blue cylinders.

clusters can survive just because the local accumulation of long monomer sequences has raised the melting point. According to eq 1, symmetric compositions favor the occurrence of demixing. For the memory effect observed in experiments, the lower limit of molar mass (1300 g/mol for G

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Figure 13. Cartoon to demonstrate the local segregation of long crystalline sequences (red stripes) remaining after melting of large crystallites, which changes the local environment and raises the melting point in the local area.

sequences remaining during annealing is demonstrated in Figure 13. In our simulations, the crystallization-induced sequence segregation is mainly driven by the different crystallizability between monomers and comonomers. The substantial trend of demixing between long and short sequences is weakened by a well distribution of sequence lengths in a homogeneous mixing state of various sequence lengths of copolymers but can be enhanced by sequence-length segregation during first-time crystallization. Interplay of phase separation and crystallization is an important issue in polymer physics.40 Here demonstrates a good example of the complicated interplay between polymer crystallization and phase separation in homogeneous random copolymer systems.

2.2% comonomer mole fraction) and the higher limit of comonomer content (4.53% comonomer mole fraction) occupy two extreme concentrations of sequence lengths required for the observation of segregation and crystallization. If the molar mass of copolymers is too low, there are too little comonomers in each polymer to make an enough amount of comonomer-rich component for microphase separation. On the other hand, if the comonomer content of copolymers is too high, the sequence length is too short to make an enough amount of long-sequence component for microphase separation. Therefore, the memory effect disappears in both extreme cases above. The comonomer contents in our simulations do not directly correspond to the same scales in real copolymers because for the formation of ordered domains in our simulations, chain folding starts around three monomers, while in reality chain folding starts around 150 carbon atoms on the chain.39 Therefore, in real copolymers, small amount of comonomers has already made a significant impact to their crystallization behaviors. In our present simulations, the critical annealing temperatures are still lower than the melting point of homopolymers, as demonstrated in Figure 2. The upper critical annealing temperature of the memory effect corresponds to the critical point of demixing for those long sequences, which in reality could be much higher than simulations due to more inhomogeneous distributions of sequence lengths (in present simulations the sequence distribution is strictly homogeneous for statistically random copolymers) as well as due to much larger sequence lengths (in present simulations the effective sequence lengths are much smaller). Thus, in real ethylenebased random copolymers, the critical annealing temperatures could be even higher than the equilibrium melting point of homopolymers.31

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ASSOCIATED CONTENT

S Supporting Information *

A movie showing the morphology evolution of one lamellar crystal in the homogeneous random copolymer of f 2 = 0.44 and chain length 128 during the annealing at Ta = 2.6 above the equilibrium melting point of the copolymer; each frame was recorded in every 200 MCcs; those crystalline bond containing more than five parallel neighbors are drawn as yellow cylinders. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (W.H.). Present Address §

M.V.: ExxonMobil Research and Engineering Company, Clinton, NJ 08801.

Notes

The authors declare no competing financial interest.

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IV. CONCLUSIONS Both experiments and molecular simulations have confirmed the strong memory effect of crystallization persisting at high annealing temperatures of homogeneous random copolymers. Such a memory effect can accelerate subsequent crystallization after annealed above the equilibrium melting points of copolymers, which is different from the conventional cases in both homopolymers and heterogeneous copolymers. By means of dynamic Monte Carlo simulations, we evidenced that the strong memory effect can be attributed to the segregation of long sequences enhanced by first-time crystallization and survived upon annealing above the equilibrium melting points of homogeneous random copolymers. This segregation changes the local concentration of monomers and more significantly, the local concentration of long sequences, resulting in a shifting-up of the local melting point to accelerate the subsequent crystallization on cooling. A cartoon to demonstrate such a local segregation of long

ACKNOWLEDGMENTS The financial support from National Natural Science Foundation of China (NSFC Grants 20825415 and 21274061) and from National Basic Research Program of China (Grant 2011CB606100) is appreciated.

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REFERENCES

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dx.doi.org/10.1021/ma400842h | Macromolecules XXXX, XXX, XXX−XXX