Monte Carlo Simulation of Strain-Enhanced Stereo-Complex Polymer

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Monte Carlo Simulation of Strain-Enhanced Stereo-Complex Polymer Crystallization Xinchao Guan, Jiping Wang, and Wenbing Hu J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b07499 • Publication Date (Web): 13 Nov 2018 Downloaded from http://pubs.acs.org on November 22, 2018

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The Journal of Physical Chemistry

Monte Carlo Simulation of Strain-Enhanced Stereo-Complex Polymer Crystallization Xinchao Guan, Jiping Wang, Wenbing Hu* Department of Polymer Science and Engineering, State Key Lab of Coordination Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210023, China *Correspondence email: [email protected]

Abstract: We performed dynamic Monte Carlo simulations to investigate the strain-induced polymer crystallization under separate enhancements of driving forces for homo-component and stereo-complex crystallization in the half-half symmetric racemic polymer blends. The results showed that polymer strain significantly enhances the stereo-complex crystallization, in comparison to the parallel cases of template-induced crystal growth without any strain in the previous simulations. We attributed the results to the strain-induced polymer crystallization favoring intermolecular crystal nucleation at high temperatures, which benefits the stereo-complex crystallization. Our observations provided a molecular-level interpretation to strain- or shear-enhanced stereo-complex crystallization in the racemic polylactide blends.

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INTRODUCTION The stereo-isomer racemic blends of poly(D-lactide) and poly(L-lactide) can form the stereo-complex crystals with alternating components of parallel-packing bonds,1,2 which hold higher modulus3 and stronger thermal resistance,1 but less harvest of crystallinity under the conventional molding process.4 Several methods have been developed, including changing the microstructures as block copolymers,5,6 nucleation agents,7 cross-linkers8 or compatibilizers,9,10 as well as changing the processing styles such as electrospinning,11,12 repeated casting13 and oscillation shear injection molding.14 Adding poly(D-lactide)-based diblock copolymers even enhances the crystallization rate of bulk poly(L-lactide) as well as its product extensibility.15 Industrial polymer processing generally performs shearing. Therefore, it is worth to pay attention to the fact that shear flow enhances stereo-complex crystallization during both non-isothermal16 and isothermal17 crystallization processes while the parallel quiescent conditions without shearing mainly harvest homo-component crystals. The stereo-complex crystallinity can be dramatically raised from 1.7% under quiescent conditions to 22% under shearing.18 In principle, it is polymer strain that enhances the formation of stereo-complex crystals in the racemic blends.19 However so far, the microscopic mechanism of strain-enhanced stereo-complex crystallization is still unclear. Stereo-complex polylactide crystals exhibit the melting point 50 degrees higher than the homo-component counterpart,1 which should provide much larger supercooling to enhance stereo-complex crystallization but the result under quiescent conditions does not behave like that.4 Under quiescent conditions, polymer crystallization intends to make chain-folded lamellar crystals,20-23 which can be attributed to the dominant intramolecular secondary crystal nucleation at the lateral growth front.24-26 In the racemic blends, polymers holding the same-specie neighbors 2


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AA or BB in the homo-component crystals favor slightly more intramolecular crystal nucleation, while polymers holding alternating-specie neighbors AB in the stereo-complex crystals intend slightly more intermolecular crystal nucleation. Dynamic Monte Carlo simulations of template-induced lamellar growth in the half-half symmetric racemic polymer blends with separate enhancements of driving forces for homo-component and stereo-complex crystals indeed have demonstrated the kinetic priority of homo-component crystals upon lamellar crystal growth.27 The situation can be changed by large polymer strain, since strain-induced polymer crystallization favors more intermolecular crystal nucleation at high temperatures, as proposed firstly by Flory28 and evidenced later on by Keller.29 Dynamic Monte Carlo simulations have demonstrated that the preferences of crystal nucleation can switch from intramolecular to intermolecular modes upon the increase of strain in bulk polymers.30 Different trends of chain-folding in the crystals during strain-induced polymer crystallization have revealed separate stages of crystal nucleation, crystal growth and strain-induced melting-recrystallization.31 Thus, many structure-related factors in strain-induced polymer crystallization have been systematically investigated, such as binary chain lengths,32 comonomers,33-35 solutions36 and blends.37 Since strain-induced crystallization favors intermolecular crystal nucleation, and the latter helps the stereo-complex crystal formation, we expect that polymer strain will enhance the stereo-complex crystallization in the half-half symmetric racemic blends. Thus, the strain- or shear-enhanced stereo-complex crystallization in racemic polylactide blends can be explained. In this report, we perform dynamic Monte Carlo simulations to demonstrate strain-enhanced stereo-complex crystallization under separate enhancements of driving forces for two kinds of crystals in a half-half racemic blend. The results will provide a further evidence for the 3



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strain-induced preference of intermolecular crystal nucleation and confirm our expectation above.

SIMULATION TECHNIQUES The traditional lattice model of polymers was employed in our dynamic Monte Carlo simulations of symmetric polymer blends with the crystalline order driven by parallel packing of the bonds.38 We set up totally 1920 chains, each occupying consecutive 128 lattice sites, in the initial lattice box of 16×128×128 (XYZ) cubic cells, for a half-half symmetric blend of two polymer chains separately denoted as A and B chains in mimic to a racemic pair of stereo-isomers, as demonstrated by the left snapshot in Figure 1. AA bonds could make parallel packing with either AA bonds or BB bonds, depending upon their separate driving forces for homo-component and stereo-complex crystallization. The polymer volume fraction was kept at 0.9375 for the bulk polymer phases, and the rest vacancy single sites were treated as free volume necessary for the micro-relaxation of polymer chains. In mimicking the micro-relaxation of real polymers (the so-called dynamic simulations), polymer chains were moving with the single monomer hopping into its vacancy neighbor similar to the well-known bond-fluctuation model, sometimes jointed with local sliding diffusion along the chain in order to maintain the chain connection.39 The unit of time period, one Monte Carlo cycle, was defined by the number of trial moves equal to total monomers in the sample system. The preset-ordered polymer chains were relaxed over 1106 Monte Carlo cycles into random coils, with two chain ends separately restricted (still mobile) in the 2D space of the parallel plates at both ends of X-axis.

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Figure 1. Snapshots of binary symmetric polymer blends before and after stretching along the X-axis to 100% strain at kT/Ec =5.1, Ea/Ec=1000 and Ep/Ec=1 for all the bond pairs, with the polymer bonds of two components shown separately in blue and yellow cylinders.

The conventional Metropolis sampling algorithm was applied to each trial micro-relaxation move for its acceptability judged by the following energy criteria:

E E E E  (a  p  p  b  a ) c kT Ec Ec kT

(1)

where a was the net number of non-collinear bonds, p was the net number of non-parallel-packed bonds, b was the net number of new vacancy sites introduced by stretching and then expelled from the central X-line, Ep was the parallel-packing potential reflecting the crystalline order of monomer bonds, Ec was the energy difference between collinear and non-collinear bonds reflecting the chain flexibility, Ea was the central-fugitive potential reflecting the normal stress upon the affine deformation of strain, k was the Boltzmann’s constant, and T was the temperature. Below we fixed the normal stress as Ea/Ec=1000 to repel the new vacancy sites as soon as possible, and the reduced temperature high enough at kT/Ec =5.1 (below briefed as T) to observe 5



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strain-induced crystallization, and split Ep/Ec into EAA, EBB and EAB for various bond pairs in two homo-components and stereo-complex crystallization, respectively. The mixing interactions between A- and B-chains were set as zero (athermal mixing) to reflect a racemic blend, which was site-site interactions between two species different from bond-bond interactions setting for crystallization between two selective species.38 The detailed procedure to realize a homogenous stretching of polymers has been reported in previous simulations of strain-induced polymer crystallization.30 Two chain ends of each polymer are separately restricted in the 2D spaces of the parallel end-plates with their spacing increased step-by-step to mimic the stretching process under a constant strain rate, as demonstrated by the right snapshot in Figure 1. Figure 1 shows the disordered state of 100% strain after step-by-step stretching (up to 16 steps expanding along X-axis), while crystallization has not yet started under this thermodynamic condition. The model represents an ideal strain situation of bulk polymers, which does not fit exactly to any physically realistic deformation in polymer flows and networks. Furthermore, since polymer chains were initially aligned with each other and all chain ends were restricted on the corresponding plates, there were no actual entanglements between polymer chains. During each step of stretching, all the polymer chains were cut by the YZ plane at a randomly selected X-coordinate. Their right parts were shifted as a whole towards one more lattice site, and all the broken segments on the cutting plane were then reconnected by performing local sliding diffusion along the rest segments in their left parts. A long-term relaxation (4000 Monte Carlo cycles) was then provided to realize a homogeneous stretching in each step, and meanwhile those new vacancy sites were removed to maintain the bulk occupation density in 0.9375. Thus, the strain rate was fixed at 6.25%/4000 Monte Carlo cycles. 6


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RESULTS AND DISCUSSION In previous simulations,27 we have compared the rates of template-induced lamellar crystal growth under separate and parallel enhancements of driving forces for homo-component and stereo-complex crystals in the racemic blends, and observed the obvious priority of homo-component crystal growth. The temperature T=5.1 was high enough to avoid any spontaneous crystallization under the enhancements of driving forces. In order for a direct comparison to the previous observations, we chose the same high temperatures for strain-induced crystallization. We firstly observed the strain-evolution curves of overall crystallinity in two sample series with the driving forces separately enhanced for two kinds of crystals with different molecular packing, i.e. homo-component AA crystals and stereo-complex AB crystals. The corresponding results are split into Figures 2a and 2b. The overall crystallinity was defined as the fraction of monomer bonds containing more than five parallel neighbors in whatever species, obtained at the end of 4000 MC cycles in each step of strain. One can see that the enhancements of both driving forces initiate the crystallization at smaller strains. We defined the onset strain of crystallization by the cross-over point of two extrapolation lines separately from the beginning and the transition regions, as for example demonstrated in Figure 2a. The results of onset strains are summarized in Figure 2c. One can see that the separate enhancements in the same extents for two kinds of crystals make almost the same onset strains for crystallization, which actually have implied the elimination of the previous priority for homo-component crystallization under quiescent conditions.

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a.

0.5

Enhancing EAA 1.10 1.15 1.20 1.25 1.30

Crystallinity

0.4

0.3

0.2

0.1

Stretching 0.0

0

100

200

300

400

Strain (%)

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Enhancing EAB 1.10 1.15 1.20 1.25 1.30

Crystallinity

0.4

0.3

0.2

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Stretching 0.0

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Strain (%)

c. Onset strain for crystallization (%)

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260

240

Enhancing EAB

220

Enhancing EAA 200

180

160 1.05

1.10

1.15

1.20

1.25

1.30

1.35

Enhancing driving force (Ep/Ec) 8


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Figure 2. Strain-evolution curves of overall crystallinity for binary polymer blends with separate enhancement of (a) EAA while keeping EAB=EBB=1.0 and (b) EAB while keeping EAA=EBB=1.0 as labeled at the temperature T=5.1. The overall crystallinity was defined by the fraction of the bonds containing more than five parallel neighbors of whatever components. (c) Onset strains for crystallization under the separate enhancement of driving forces for EAA and EAB. The onset strains were recorded from the cross-over point demonstrated for example in (a). The results were averaged over three independent simulations.

One can look into more details of compositions in the overall crystallinity of strain-induced crystallization under the separate enhancements of driving forces, for instance, 1.30. We defined the AA crystallinity by counting those crystalline bonds (again in whatever species) surrounded by more than 3/5 fractions of A-bonds , and defined the AB crystallinity by counting those crystalline bonds surrounded by 2/5 to 3/5 fractions of A-bonds. The strain evolution curves of AA crystallinity and AB crystallinity are compared to the overall crystallinity as shown in Figures 3a and 3b, separately for the enhancement cases of EAA=1.30 and EAB=1.30. One can clearly see that in comparison of two separate enhancements of driving forces, the dominant contributions in the overall crystallinity are accordingly switched from homo-component AA crystals to stereo-complex AB crystals, although they exhibit the similar onset strains of crystallization during stretching.

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a.

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Enhancing EAB=1.30 0.4

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Overall 0.3

0.2

AB AA

0.1

Stretching 0.0

0

100

200

300

400

Strain (%) Figure 3. Strain-evolution curves of overall, AA and AB crystallinity for binary polymer blends with separate enhancements of (a) EAA=1.30 while keeping EAB=EBB=1.0 and (b) EAB=1.30 while keeping EAA=EBB=1.0 at temperature T=5.1.

The kinetics of isothermal crystallization at high temperatures appears as more sensitive to the thermodynamic driving forces than the non-isothermal crystallization kinetics in a cooling 10


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process. The similar situation is true for the stretching process. We therefore performed the isothermal crystallization process for 1106 Monte Carlo cycles by holding the sample at the constant strains of 200% and the fixed temperature T=5.1, under separate enhancements of two driving forces. During the isothermal and iso-strain crystallization, we recorded the half crystallization time by tracing the time evolution of overall crystallinity, as for example, demonstrated in Figure 4. In Figure 4, the dashed lines were separately extrapolated from the transition region and the saturate region to get the final point of crystallization Cf, and the half-crystallization time was recorded by the middle point between Cf and the initial point Ci.

0.50

Cf

0.45 0.40

Crystallinity

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.35

(Ci+Cf)/2

0.30 0.25 0.20 0.15 0

Ci

t1/2 200000

400000

600000

800000

1000000

Time (Monte Carlo Cycles) Figure 4. Time evolution curve of crystallinity under the enhancement of driving forces for homo-component crystal EAA=1.30 while keeping EAB=EBB=1.0 and holding 200% strain at the temperature T=5.1. The half crystallization time was recorded at the middle point between the initial and the final crystallinity (the final was the cross-over of two extrapolation lines from the transition and the saturate regions). 11



The reciprocal half crystallization times represent the crystallization rates under separate enhancements of driving forces, as the results summarized in Figure 5. One can see that the parallel enhancements of the driving forces separate for homo-component AA and stereo-complex AB crystallization under 200% strain and T=5.1 will make the stereo-complex crystallization much faster than homo-component crystallization. Such a trend is right opposite to the linear crystal growth rates under the quiescent conditions without any strain at T=5.1, where the homo-component crystals grow much faster than the stereo-complex crystals under the parallel enhancements of the driving forces,27 as the inset figure shown in Figure 5.

18 16

0.0016

Linear growth rate

20

Reciprocal time (10-6 MC cycle-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Quiescent condition

EAB

0.0012

0.0008

14

EAA

0.0004

12 10

EAB

0.0000 1.10

1.15

1.20

1.25

Driving force (E)/ Ep/Ec

EAA

1.30

8 6

Strain=200%

4 2

1.10

1.15

1.20

1.25

1.30

Enhancement of driving force (Ep/Ec) Figure 5. Reciprocal half crystallization times of isothermal crystallization at various enhanced driving forces separate for homo- and stereo-complex crystals under 200% strain and the other parameters same as in Figure 2. The data were averaged over three independent simulations. The inset figure were the parallel-comparing results for the linear crystal growth rates reported in 12


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Reference 27.

With the enhancement of driving forces from 1.20 to 1.25 in Figure 5, there appears a jump of increase in the crystallization rates. This may be attributed to the saturation of the data at 1.30, which could be lowered by fast crystallization due to the difficulty to read properly the initial point for the estimation of half-crystallization time. The visual inspection on the final crystalline states under the separate enhancements of two driving forces confirmed the corresponding harvests of AA and AB crystallites during isothermal crystallization, as the snapshots shown in Figure 6. In the left snapshot of Figure 6, larger AA domains in the case of EAA=1.30 can be attributed to the enhanced homo-component AA crystallization. Apparently, the existing of 200% strain changes the priority sequences of crystallization from homo-component crystals to stereo-complex crystals. Such an effect clearly demonstrates the favorite intermolecular crystal nucleation in strain-induced polymer crystallization at high temperatures, which benefits the stereo-complex crystallization. Therefore, we provided the molecular-level scenario for the microscopic mechanism of shear- or stain-enhanced stereo-complex crystallization in the racemic blends of polylactides.

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Figure 6. Snapshots comparing the final states of separate enhancements of driving forces for AA and AB crystals at 1.30 during isothermal crystallization at constant 200% strain with crystalline A-bonds showing in blue cylinders and crystalline B-bonds in yellow cylinders. The other energy parameters are the same as in Figure 2.

CONCLUSION By means of dynamic Monte Carlo simulations, we observed the priority of stereo-complex crystallization during strain-induced crystallization in the half-half symmetric blends of racemic polymers, in contrast to the previous simulation results under quiescent conditions. We attributed the priority of stereo-complex crystallization to the preferences of intermolecular crystal nucleation upon strain-induced polymer crystallization at high temperatures. Therefore, our present results on the one hand made a further evidence for the switching of preferred crystal nucleation from intramolecular to intermolecular modes upon the stretching of polymers, on the other hand offered a molecular interpretation to the shear- or strain-enhanced stereo-complex 14


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crystallization in the racemic polylactide blends. The latter could be practically meaningful for the industrial processing of the biodegradable materials.

ACKNOWLEDGEMENTS The financial supports from National Natural Science Foundation of China (NO. 21474050 and NO. 21734005) and the Program for Changjiang Scholars and Innovative Research Teams in University are gratefully acknowledged.

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TOC Graphic

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Monte Carlo Simulation of Strain-Enhanced Stereo-Complex Polymer

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