Stereocomplex Crystallization in Asymmetric Diblock Copolymers Studied by Dynamic Monte Carlo Simulations

Stereocomplex crystallization in asymmetric diblock copolymers was studied using dynamic Monte Carlo simulations, and the key factor dominating the formation of stereocomplex crystallites (SCs) was uncovered. The asymmetric diblock copolymers with higher degree of asymmetry exhibit larger difference between volume fractions of beads of different blocks, and local miscibility between different kinds of beads is lower, leading to lower SC content. To minimize the interference from volume fraction of beads, the SC formation in blends of asymmetric diblock copolymers was also studied. For the cases where the volume fractions of beads of different blocks are the same, similar local miscibility between beads of different blocks and similar SC content was observed. These findings indicate that the volume fraction of beads of different blocks is a key factor controlling the SC formation in the asymmetric diblock copolymers. The SC content can be regulated by adjusting the difference between the contents of beads of different blocks in asymmetric diblock copolymers.


INTRODUCTION
The development of degradable plastics based on renewable resources is a hot topic in the field of polymer materials and engineering. [1−5] On the one hand, the application of these degradable polymeric products is conducive to environmental protection, since the "white pollution" caused by non-degradable plastics can be avoided. On the other hand, using polymeric materials generated from renewable resources can reduce the dependence of industry on petroleum, and this is especially important for countries lacking natural resources. Among these degradable plastics based on renewable resources, poly(lactic acid) (PLA) is one of the most concerned materials, [3−7] and it has been used to manufacture a number of products, such as surgical sutures, membranes and fibers.
However, it should be noted that PLA still cannot totally replace the conventional plastics, owing to its relatively weak mechanical properties and low melting point. [8−10] Ikada et al. first observed that stereocomplex crystallites (SCs) can form between poly(L-lactic acid) (PLLA) and poly(D-lactic acid) (PDLA) chains during crystallization. [5] These SCs have much higher melting point compared with that of homocrystallites (HCs) formed in neat PLLA. [11,12] In this case, the presence of the SCs in PLLA/PDLA blends can effectively improve thermal stability of PLA materials. However, in PLA blends with high molecular weights, both SCs and HCs form together during crystallization, [1,4,13,14] and the increase of molecular weights of PLA can lead to the decrease of SC content. [15−17] Accordingly, nowadays, the key point of the research on the improvement of mechanical properties and thermal stability of PLA materials becomes the enhancement of the SC formation in PLA blends. [3,18] Some useful methods for promoting the SC formation have been proposed, such as stretching or shearing PLA chains, [19−23] synthesizing PLLA-PDLA block copolymers or star-shaped polymers [24−28] and adding some new components that can interact with both PLLA and PDLA chains. [9,29,30] These excellent studies are mainly based on two mechanisms to control the SC formation: the change of nucleation/crystallization mode from intramolecular chain-folding to intermolecular packing and the improvement in local miscibility between PLLA and PDLA chain segments. [22,29,31−36] For instance, Hu et al. used dynamic Monte Carlo (MC) simulations to explore the microscopic mechanisms of strain-enhanced stereocomplex formation in polymer blends, and detected that the improvement of the SC formation could be attributed to intermolecular packing of highly deformed chain segments during crystallization. [22] Our former simulation work also showed that the increase of the grafting density of polymer chains grafted on substrate can cause the raise of chain extension and the dominance of crystallization mode of intermolecular packing during crystallization, thus promoting the appearance of more SCs. [32] In addition, compared with linear PLA chains, PLLA-block-PDLA copolymers have higher local miscibility and therefore stronger ability of SC formation. [24][25][26][27]34,37] For block copolymers, it can be speculated that the distribution of PLLA and PDLA blocks in chains should affect the SC formation. Pan et al. reported that symmetric PLLA-PDLA diblock copolymers exhibit stronger ability of stereocomplexation than the corresponding asymmetric diblock copolymers. [24] Unfortunately, the corresponding research work focusing on the effect of the asymmetry of diblock copolymers on the SC formation is still lacking, and more work is urgently needed.
Previously, Hu's group and our group both demonstrated that dynamic MC simulations provide a very powerful research method for investigating the underlying mechanisms of polymer crystallization, [38−53] including the SC formation. [17,22,31,32,34,54] Thus, in the current simulation work, we further applied dynamic MC simulation to study the effect of asymmetry of diblock copolymers on the SC formation. It was detected that the SC content decreases with the increaseing degree of the asymmetry of diblock copolymers, which can be mainly attributed to the reduction in the local miscibility between segments belonging to different blocks caused by the increase of the difference between the contents of segments of different blocks.

SIMULATION DETAILS
Our current simulations were carried out on a lattice, and the corresponding simulation box contained 32×32×32 lattice sites in total. In the box, polymer beads could only exist on the lattice points. In this case, polymer chains used in the present simulations could be regarded as lattice chains (one kind of coarse-grained polymer chain model). [46] Based on the current lattice model, one bond formed by two consecutive beads had 26 possible orientations, that is, orientations along 6 axes, 8 body diagonals and 12 face diagonals. Then, the bond length of the lattice chains could be 1, and . In order to reveal the effect of the asymmetry of diblock copolymers, 240 diblock copolymer chains with chain length of 128 beads but different degrees of asymmetry were introduced. Since the radius of gyration of random chain coils with chain length of 128 beads was around 12 lattice sites (smaller than the side length of the box (32 lattice sites)), no confinement effect existed in our current simulation systems. The diblock copolymers were marked as A-B diblock copolymers. The volume fraction of A beads in one diblock copolymer chain (f a ) was 0.5, 0.25, 0.125 and 0.0625, respectively, as illustrated in Fig. 1 (the corresponding volume fraction of A beads in these copolymer systems was also 0.5, 0.25, 0.125 and 0.0625, respectively). Then, the A-B diblock copolymer with f a =0.5 was symmetric, while the other diblock copolymers with f a =0.25, 0.125 and 0.0625 belonged to the asymmetric diblock copolymers. The bead density of these copolymers was 0.9375, thus these systems could be considered as the bulk ones. In addition, we also established the systems of the blends of two asymmetric diblock copolymers with the complementary volume fractions of A beads, that is, the volume fraction of A beads in the first copolymer chain (f a1 )/that in the second copolymer chain (f a2 )=0.25/0.75, 0.125/0.875 and 0.0625/0.9375, respectively. In the blends, the volume fractions of both the A and B beads were 0.5.
In order to achieve the formation of SCs, we should first understand the underlying mechanism inducing the appearance of SCs in PLA blends. Using infrared spectroscopy, Zhang et al. found that intermolecular hydrogen bonds can form between neighbouring PLLA and PDLA segments during crystallization in their blends, [55] while no similar hydrogen bonds appear among neat PLLA or PDLA system. The formation of the hydrogen bonds in SCs can induce the appearance of higher energy reduction during crystallization and then higher thermal stability compared with HCs without the hydrogen bonds. Hu et al. firstly introduced a parameter for parallel interaction (E p ) to reflect the variation of compact packing energy of adjacent bonds. [38,46] Apparently, the formation of the hydrogen bonds between adjacent PLLA and PDLA segments can result in the higher reduction of packing energy when the parallel packing of one PLLA bond and one PDLA bond occurs, compared to the energy change caused by the parallel packing of two PLLA bonds or two PDLA bonds. According to this phenomenon, in the simulations we should set a higher value of E p for the packing of one A bond and one B bond. Hu et al. have found that the presence of a higher value of E p for two parallel bonds of different types can promote the stereocomplex crystallization of polymer blends. [22,31] Accordingly, on the basis of their pioneer simulation work, [22,31] different values of E p were set for the parallel bonds with different types. For chain motions on the lattice, we adopted the micro-relaxation mode of chains. Then, chain segments could move based on the single-site jumping or chain segment sliding. We introduced periodic boundary conditions in the simulations. During simulation, whether the attempts of chain segment movements could be accepted or not was judged by the conventional Metropolis sampling algorithm. Each polymer bead had one opportunity to move at one MC cycle. In this condition, the simulation time could be reflected by the number of the MC cycles. The following equation was used to obtain the change of potential energy (ΔE) owing to chain segment motions: where E c was the change in energy correlated with the noncollinear connection of two consecutive polymer bonds, E p1 was the change in energy correlated with two nonparallel packed bonds both belonging to A polymer (that is, A-A bonds) in neighboring regions, E p2 was the change in energy correlated with two nonparallel packed bonds both belonging to B polymer (that is, B-B bonds) in neighboring regions (E p1 and E p2 corresponded to driving force for HC formation), E p3 was the change in energy correlated with two nonparallel packed bonds of different types (A-A bond and B-B bond) in neighboring regions, which corresponded to driving force for SC formation, E p4 was the change in energy correlated with two nonparallel packed bonds, one of which was A-B bond (that is, the block junction bond), Δc was the change in number of pairs of bonds that were not collinear, Δp 1 was the change in number of pairs of A-A bonds that were not parallel packed, Δp 2 was the change in number of pairs of B-B bonds that were not parallel packed, Δp 3 was the change in number of bond pairs containing one A-A bond and one B-B bond that were not parallel packed, and Δp 4 was the change in number of non-parallel packed bond pairs containing at least one A-B bond. As mentioned above, different values of the parameter for parallel interaction were set for the packing of two A-A bonds (E p1 /E c =1) or two B-B bonds (E p2 /E c =1), the packing of one A bond and one B bond (E p3 /E c =1.2) and the packing of two bonds containing at least one A-B bond (E p4 /E c =0), as shown in Fig. 2. Since the value of E p4 /E c was set as 0, the block junction bonds of the block copolymers could not participate in crystallization. The similar parameter settings were also used in our former simulations on the SC formation in the mixed polymers grafted on a substrate, the multi-block copolymers and the polymer blends filled with two-dimensional nanofillers. [32,34,54] In addition, we used lattice chain model in the simulations instead of the real PLLA and PDLA chains. It should be noted that the lattice chain model cannot be used to directly reflect the real PLA chains. Differently, for molecular dynamics simulations, the all-atom model, the united-atom model and some coarse-grained models can be used to directly reflect the real polymer chains. In this condition, the crystallization mechanisms of some special polymers can be successfully uncovered. In the current study, based on this lattice chain model, we focused on the universal mechanisms of stereocomplex crystallization. Firstly, the copolymer systems were relaxed in an athermal condition for 10 6 MC cycles in order to gain an equilibrium random coil state, which was then used as the initial state for subsequent crystallization. The subsequent isothermal crystallization took place at the reduced temperature (kT/E c ) of 3.0. In the following sections, kT/E c was marked as T * for convenience. Fig. 3 shows the crystallinity evolutions for the different diblock copolymers including the symmetric and asymmetric diblock copolymers during isothermal crystallization at T * =3.0. Here, crystallinity is treated as the ratio of the number of bonds having more than 5 parallel neighboring bonds to the total number of bonds in the systems. [46] Besides, this bond will be regarded as a crystalline bond. It can be clearly seen that during crystallization the values of the crystallinity for all these diblock copolymer systems increase with increasing simulation time, indicating the occurrence of crystallization in these diblock copolymers. In addition, all these different diblock copolymers exhibit similar crystallization rates, attributed to the same crystallizability of the A and B blocks. Furthermore, we try to reveal the effect of the asymmetry of the diblock copolymers on the formation of SCs. In the present work, we still use the same definitions of SC and HC bonds as those used in our former simulation work. [32,34,54] When in the neighboring domain of one crystalline bond (A/B crystalline bond), the values of the ratio of the amount of A/B crystalline bonds to that of B/A crystalline bonds are between 0.5 and 2, this bond would be defined as one SC bond. Otherwise, when the values of this ratio are larger than 2 or smaller than 0.5, and this bond would be defined as one HC bond. [32,34,54] Our definition of SC bond was inspired by the experimental results of Tashiro et al., which revealed that in SCs the ratio of the amount of PLLA crystalline segments to that of PDLA crystalline segments is within a certain range. [56] Then, to understand the difference between the two kinds of crystalline bonds (that is, HC and SC bonds) more intuitively, the snapshots of the morphologies of the HCs and the SCs in the diblock copolymers with the different values of f a are shown in Fig. 4. In the figure the crystalline bonds belonging to the different kinds of blocks are drawn with different colors. Obviously, for the HCs, the crystalline bonds belonging to the same kind of blocks (the cylinders with same color in Fig. 4) are parallel with each other in local domains. On the contrary, for the SCs, the crystalline bonds of the different kinds of blocks (the cylinders with different colors in Fig. 4) are aligned in parallel in local regions. In addition, it can also be seen in Fig. 4 that the number of the HC bonds increases with the decrease of the values of f a , while that of the SC bonds decreases, indicating that the enhancement of the asymmetry of the diblock copolymers leads to the significant decline in the final SC content. In experiments, Pan et al. also observed that the formation ability of SCs in the asymmetric PLLA-PDLA diblock copolymers is weaker than that in the symmetric block copolymers. [24] However, the detailed underlying mechanism of this phenomenon is still unclear.

Asymmetric Block Copolymers
To quantitatively reveal the dependence of the SC content on the asymmetry of the diblock copolymers, we further count the final SC fraction as a function of the values of f a . Here, we define the final SC fraction as the value of the number of SC bonds divided by the total number of crystalline bonds at 2.0×10 5 MC cycles. As shown in Fig. 5, it can be seen that as the values of f a decrease, the final SC fraction is also reduced, suggesting that the increase of the degree of the copolymer asymmetry can apparently weaken the formation ability of SCs. In addition, it can also be seen that the values of the final SC fraction linearly depend on the values of f a , implying that the volume fraction of beads of different blocks or the degree of asymmetry is the key factor dominating the formation of SCs in the asymmetric diblock copolymers. Subsequently, we further focus on the mechanism controlling the formation of SCs in the asymmetric diblock copolymers. As revealed in the previous corresponding experiment and simulation work, there are two main mechanisms controlling the formation of SCs, i.e., the local miscibility of different polymer chain segments and the nucleation mode. [22,29,31−35] Fig. 6(a) depicts the evolutions of mixing parameters of B beads during crystallization for these different diblock copolymer systems. The mixing parameter of B beads was defined as the fraction of A beads existing in the local neighboring regions of B beads. [54] This mixing parameter of B beads can be used to reflect the local miscibility between A and B beads. As shown in Fig. 6(a), the diblock copolymers with the lower values of f a exhibit the lower mixing parameters of B beads during crystallization process. For the diblock copolymer with a lower value of f a (that is, the higher degree of asymmetry), the fraction of A beads is lower. In this condition, the possibility of B beads to meet A beads in the corresponding neighboring domains will be lower, thus resulting in the lower values of the mixing parameters of B beads. In short, the systems with the higher degree of asymmetry have weaker local miscibility, which is not conducive to the SC formation. Fig. 6(b) further displays the dependence of the mixing parameters of B beads in the initial state of the different block copolymers on the values of f a . It can be seen that there is a linear relationship between the mixing parameters of B beads in the initial state and the values of f a , demonstrating that the local miscibility is closely dependent on the fraction of different beads. It should be noted that, if the value of f a is improved from 0.5, the diblock copolymer will also become asymmetric. Then, the systems with f a >0.5 (asymmetric diblock copolymers with higher volume fraction of A block) will also have lower mixing parameters and thus lower fraction of SCs compared with that with f a =0.5 (symmetric diblock copolymers).
Moreover, we go further to examine whether the change of asymmetry of block copolymers will affect the nucleation mode. Fig. 7 depicts the adjacent chain-folding probability in crystals during crystallization at T * =3.0 for these diblock copolymers with different values of f a . The definition of the adjacent chain-folding probability is the same as that used in the former work. [39,40] Namely, the adjacent chain-folding probab-ility in crystals is defined as the fraction of crystalline stems in crystals that are directly connected by short loops composed of less than five beads. [32,38−40] Generally, the value of the adjacent chain-folding probability can be used to reveal the nucleation or crystallization mode of polymers. [32,38−40] As depicted in Fig. 7, it can be found that all the block copolymers  exhibit the similar evolutions of the adjacent chain-folding probability due to the same chain length of these different diblock copolymers. In other words, the nucleation mode in the diblock copolymers with different asymmetry is not the mechanism affecting the formation of SCs. In the current systems, only the local miscibility between different blocks plays the important role in the SC formation.

Blends of Asymmetric Block Copolymers
In the above diblock copolymer systems, the diblock copolymers with higher degree of asymmetry (that is, smaller value of f a ) have lower volume fraction of A beads. In this condition, it is difficult to directly extract the influence of asymmetry on the SC formation without considering the influence of the fraction of beads. Thus, in order to reveal the effect of the asymmetry of diblock copolymers on the SC formation excluding interference from the volume fraction of beads, we further investigate the SC formation in the blends of the asymmetric diblock copolymers. In all the blends of the asymmetric diblock copolymers, the volume fractions of A and B beads are both fixed at 0.5, while the degree of asymmetry is different (f a1 /f a2 =0.25/0.75, 0.125/ 0.875 and 0.0625/0.9375, respectively). Then, the effect of the asymmetry of diblock copolymers can be revealed individually. As shown in Fig. 8, the crystallinity values of all the blends of the asymmetric diblock copolymers increase during crystallization, demonstrating that crystallization occurs in all the corresponding systems. Fig. 9 displays the dependence of the final SC fraction on the values of f a1 . It can be observed that the value of the final SC fraction is almost independent of f a1 , indicating that the degree of asymmetry of diblock copolymers is not the key factor affecting the SC formation. As shown in Fig. 10, the mixing parameters of B beads in the different blends of the asymmetric diblock copolymers are almost the same. In other words, these blends of the asymmetric diblock copolymers have the same volume fraction of beads, and then exhibit the similar local miscibility.
Based on these observations, it can be concluded that the volume fraction of beads belonging to different blocks is the key factor controlling the SC formation in the asymmetric block copolymers. In the asymmetric block copolymers with higher degree of asymmetry, the difference between the volume fractions of the two beads belonging to different blocks will be greater, resulting in the lower local miscibility and thus the smaller values of the final SC content. However, in the blends of the asymmetric block copolymers, although the degree of the asymmetry is still different, the volume fraction of the different beads is the same, leading to the similar local miscibility and also the similar values of the final SC content.
In addition, compared with the asymmetric blends, the asymmetric diblock copolymers should have higher local miscibility due to the presence of the block junction bonds. Then, the asymmetric diblock copolymers should have higher ability of the SC formation.

CONCLUSIONS
In the current simulations, we investigate the SC formation in the asymmetric diblock copolymers, and reveal that the key factor dominating the SC formation is the volume fraction of beads belonging to different blocks. In the asymmetric diblock copolymers, the copolymers with the higher degree of asymmetry have larger difference between the volume fractions of beads belonging to different blocks, and then the local miscibility between the beads of different blocks would be lower, leading to the lower values of the final SC content.
In the blends of the asymmetric block copolymers, the volume fraction of beads belonging to different blocks is the same, then resulting in the similar local miscibility between beads of different blocks and also the similar values of the final SC content. The current new simulation results can supply some theoretical guidance for the regulation of the content of SCs in block copolymers.

DATA AVAILABILITY
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.