Intramolecular Crystal Nucleation Favored by Polymer Crystallization

Subscriber access provided by UNIV OF NEBRASKA - LINCOLN

Article

Intramolecular Crystal Nucleation Favored by Polymer Crystallization: A Monte Carlo Simulation Evidence Rong Zhang, Liyun Zha, and Wenbing Hu J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b01757 • Publication Date (Web): 14 Jun 2016 Downloaded from http://pubs.acs.org on June 15, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry B is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 27

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

The Journal of Physical Chemistry

Intramolecular Crystal Nucleation Favored by Polymer Crystallization: A Monte Carlo Simulation Evidence

Rong Zhang, Liyun Zha, Wenbing Hu* Department of Polymer Science and Engineering, State Key Laboratory of Coordination Chemistry, Collaborative Innovation Center of Chemistry for Life Sciences, School of Chemistry and Chemical Engineering, Nanjing University, 210093 Nanjing, China *E-mail address: [email protected]; Phone: 0086-25-89686667

ABSTRACT: We performed dynamic Monte Carlo simulations of half-half binary blends of symmetric (double and mutual) crystallizable polymers. We separately enhanced the driving forces for polymer-uniform crystals and for polymer-staggered crystals. Under parallel enhancements, polymer-uniform crystals exhibit faster nucleation and growth with more chain-folding and less lamellar thickening than polymer-staggered crystals. We attributed the results to the intramolecular crystal nucleation ruined by the enhanced polymer-staggered crystallization. Our observations provide a direct molecular-level evidence to intramolecular crystal nucleation favored by polymer crystallization in quiescent solutions and melt, which yields chain-folding for the characteristic beta-sheet or lamellar morphology of macromolecular crystals. Keywords: Chain-folding; Lamellar crystals; Blends; Stereo-complex crystallization 1

ACS Paragon Plus Environment


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

Page 2 of 27

I. INTRODUCTION Crystalline stems of chain-like macromolecules exhibit a preference of adjacent folding upon crystallization from quiescent solutions or melt, resulting in the characteristic beta-sheet or lamellar morphology of crystallites.1,

2

At the lateral

growth front of polymer crystals, new crystalline stems can be sourced from either the same polymers via adjacent chain-folding or the different polymers via parallel alignment, separately corresponding to the intramolecular chain-folding and the intermolecular fringed-micelle modes of secondary nucleation.3 Zachman et al. have estimated that the chain-folded crystallites contain a lower surface free energy density on the stem-end surfaces than the fringed-micelle crystallites.4 So intramolecular crystal nucleation holds a relatively lower free energy barrier and thus becomes the kinetic preference upon crystal growth, which appears as responsible for the dominant chain-folding in the characteristically lamellar crystals.5 To reach this conclusion, however, one has to clarify two points: the origin of chain folding, and why major new crystalline stems are subject to this mode of secondary nucleation. Up to now, the intramolecular crystal nucleation model has been proposed and been demonstrated on the basis of dynamic Monte Carlo simulations of single-polymer crystallization.6 The model suggests that every new crystalline stem has been involved into an event of intramolecular secondary nucleation at the lateral growth front of lamellar crystals.6 Such an event is dominated by the free energy change in a single polymer that naturally chooses adjacent folding to minimize its lateral and stem-end exposure for a lower nucleation barrier and meanwhile to maximize its internal 2


Page 3 of 27

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


packing for a larger crystalline enthalpy. The model thus explains the kinetic origin to chain-folding. Such an origin was missing in the traditional Lauritzen-Hoffman model that took chain-folding as a pre-requisite condition.7-9 The kinetic preference of intramolecular crystal nucleation upon crystal growth has been found beneficial to understand many unique phenomena of polymer crystallization, including integer-number chain-folding of short polymers,5,

10

molecular segregation11 and

co-crystallization12, 13 in the binary blends of two chain lengths, the regime behavior of linear growth rates at high temperatures,14 a kinetic symmetry between crystal melting and growth,15 a power-law heating-rate dependence of superheating upon zero-entropy-production melting of lamellar crystals,16 the kinetic origin of semi-crystalline texture,17 and the switching between two nucleation modes upon strain-induced crystallization.18, 19 So far, a direct molecular-level evidence for the kinetic preference of intramolecular crystal nucleation upon polymer crystallization has not yet been achieved. The direct evidence should not be on the basis of one or two isolated observations on the nucleation events at the growth front, but rather on the basis of statistics over many events that result in crystal growth. To this end, a proper design on our observations of polymer crystal growth was demanded. Recently, stereo-complex crystallization in the racemic binary blends of left-handed and right-handed poly(lactic acid)s has attracted broad attention. At normal crystallization temperatures it appears as less competitive than molecular-uniform crystallization, although it contains a melting point 50 ºC higher for a larger supercooling.20 One potential mechanism for such a slow crystallization could be the 3



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

Page 4 of 27

alternating components selected at the growth front of the molecular-staggered crystals, not favorable for intramolecular crystal nucleation. In an ideal case, polymer-staggered crystals can perform intramolecular crystal nucleation by making (110) chain-folding alignment and meanwhile alternating the components layer-by-layer, as illustrated in Figure 1, under the ideal procedure same as the growth of molecular-uniform crystals. The alignment of fold-end orientations on the corresponding (110) growth fronts has been proved by four sectors of chain-folding in the single lamellar crystals.21 However, the actual events of intramolecular crystal nucleation will be ruined by the enhancement of driving forces for polymer-staggered crystallization. Therefore, a slower crystallization of molecular-staggered crystals than that of molecular-uniform crystals under parallel enhancements of driving forces can provide us a direct evidence for the kinetic preference of intramolecular crystal nucleation upon polymer crystal growth.

Figure 1. Illustration of molecular-uniform and molecular-staggered crystals in the binary blends of symmetric (double and mutual) crystallizable polymers.

Inspired from the consideration above, we designed dynamic Monte Carlo 4


Page 5 of 27

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


simulations of binary polymer blends performing both molecular-uniform and molecular-staggered crystallization. We separately enhanced the corresponding driving forces for two crystallizations, and compared the results under parallel situations. The results will demonstrate that, due to their much more favorites for intramolecular crystal nucleation, molecular-uniform crystals hold a larger amount of chain folding and exhibit faster nucleation and growth than the parallel-enhanced molecular-staggered crystals. In this approach, we obtain a direct evidence for the kinetic preference of intramolecular crystal nucleation during polymer crystallization from quiescent states.

II. SIMULATION TECHNIQUES Our Monte Carlo simulations employed the classical lattice model of symmetric polymer blends. We set up two kinds of polymer chains separately denoted as A- and B-chains, each chain occupying consecutive 128 lattice sites, in the box of 64×64×64 cubic lattice cells with periodic boundary conditions. A- and B-chains were half-half blending and the total chain number was 1920, which was subjected to the polymer volume fraction 0.9375 in mimic to the melt state. The rest 0.0625 volume fraction of singly vacancy lattice sites were regarded as free volume for the trial moves of polymer chains. The micro-relaxation model of the trial moves allowed single-site jumping and sometimes partial sliding diffusion along the chain.22 Double occupation and bond crossing were forbidden on account of the excluded volume of polymers. The bonds were oriented either along the lattice axes or along the body and face 5



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

Page 6 of 27

diagonals, so the coordination number of each bond was as high as 26. The conventional Metropolis sampling algorithm was applied to each trial move with the total potential change as given by22

E E E E pE p  cEc   s(i ) E f   [ p p  c   s (i ) f ] c (1) kT kT Ec Ec kT Here, Ep represented the parallel-packing interactions between neighboring bonds, reflecting the thermodynamic driving force for crystallization, and p was the net change of nonparallel pairs of neighboring bonds; Ec represented the collinear connection energy for consecutive bonds along the chain, reflecting the chain semi-flexibility, and c was the net change of non-collinear pairs of consecutive bonds; Ef described the frictional barrier for sliding diffusion of each bond from its parallel

neighbor, and s(i) was the number of parallel neighbors of the i-th bond along the local sliding path; k was Boltzmann’s constant and T temperature. The two kinds of chains were crystallizable with themselves as well as with each other. For simplicity we set their mutual mixing as athermal. We labeled Ep/Ec with EAA, EBB and EAB separately for two molecular-uniform and one molecular-staggered

crystals. If we set all equal driving forces, the crystallization of blends will behave the same as that of homopolymer melt. In practice, we stepwise raised EAA from 1.1 to 1.3 to enhance the driving forces for crystallization of molecular-uniform crystals, while both EBB and EAB were kept at 1. In parallel comparisons, we stepwise raised EAB from 1.1 to 1.3 to enhance the driving forces for crystallization of molecular-staggered crystals, while both EAA and EBB were kept at 1. We set Ef/Ec = 0 to allow crystal thickening and 0.3 to prohibit it. We chose the reduced temperature kT/Ec = 5.1 to 6


Page 7 of 27

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


observe the enhanced nucleation and crystal growth when Ef/Ec = 0, and 5.9 when Ef/Ec = 0.3. The time unit of our simulations was defined as Monte Carlo cycle (MCc)

that was the number of trial moves equal to the total number of monomers when all the monomers got one chance to move on average. In the following, we will first relax all the preset ordered chains to their random-coil states under athermal conditions for 106 MCcs, and observe thermal fluctuations for primary crystal nucleation in the homogeneous melt state; we then keep an array of 8th folded A-chains spanning over Y-axis at X=1 and Z=24 during athermal relaxation, and observe the relaxed polymers performing crystal growth induced by the template.

III. RESULTS AND DISCUSSION A. Crystal nucleation. We start our comparisons from the probabilities of primary

crystal nucleation between molecular uniform and staggered crystals under separate enhancement of driving forces in the symmetric blends. It is well-known that crystal nucleation is initiated by thermal fluctuations, and the maximum crystalline clusters hold the highest probability to get over the nucleation barrier and to survive as crystal nuclei. Therefore, we calculated the size distribution of crystalline clusters, and traced the time evolution of maximum crystalline clusters under separate enhancement of driving forces for two crystallization. Larger EAA would help the formation of A-A molecular-uniform crystalline clusters, while larger EAB could be beneficial to the formation of molecular-staggered crystalline clusters. Under these two parallel 7



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

Page 8 of 27

situations of enhancement, which situation exhibits more acceleration of crystal nucleation will reveal the kinetic preference between two kinds of nucleation modes. Figure 2 shows a typical result of parallel comparison between the case of EAA = 1.3 and the case of EAB = 1.3 on the time evolution curves of the sizes of maximum molecular-uniform and molecular-staggered crystalline clusters. The size of maximum crystalline cluster was defined by the maximum number of crystalline bonds aligned with each other in the single cluster among various clusters, while the crystalline bond was defined as the bond containing more than five parallel neighbors irrespective of A or B bonds. When the maximum cluster contains the ratio of NA (the number of crystalline A bonds in the maximum cluster) to NB (the crystalline B bonds in the same cluster) larger than 2, we counted the cluster as the molecular-uniform crystalline cluster. When the ratio of NA to NB became larger than 0.5 but smaller than 2, we counted the cluster as the molecular-staggered crystalline cluster. One can see that right after 7×104 MCcs, the maximum molecular-uniform crystalline clusters suddenly exceed a threshold size and raise up for crystal growth, while in parallel situation the maximum molecular-staggered crystalline clusters remain at the low level of sizes upon thermal fluctuations. Obviously, enhancing EAA appears as more effective to accelerate primary nucleation of molecular-uniform crystals than the same extent of enhancing EAB for that of molecular-staggered crystals, implying the kinetic preference of primary crystal nucleation for the molecular-uniform crystals.

8


Page 9 of 27

4

14

Size of maximum cluster /10

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


EAA=1.3, EBB=1, EAB=1

12

EAA=1, EBB=1, EAB=1.3

10 8 6 4 2 0 0

50

100

150

200

3

Time period /10 MCcs

Figure 2. Time evolution curves of maximum cluster sizes for molecular-uniform and

staggered crystalline clusters with the driving forces for three crystals of A-A, B-B and A-B as labeled when Ef/Ec = 0 and kT/Ec = 5.1. For the details of definition, see the text.

One may argue that the observation of primary crystal nucleation in Figure 2 still belongs to an isolated stochastic event. For a statistical observation on thermal fluctuations, we chose EAA and EAB values slightly smaller than 1.3 to avoid spontaneous crystal nucleation in the time window of our observations. Since the maximum clusters represent the first candidate of survivors for nuclei, Figure 3 compares the size distributions between two kinds of maximum crystalline clusters under parallel enhancements of driving forces. One can see that beyond the size 15, molecular-uniform crystalline clusters appear as much more than molecular-staggered crystalline clusters, confirming again the kinetic preference of primary nucleation for 9



molecular-uniform crystals. 400

Number of maximum clusters

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

Page 10 of 27

EAA=1.27, EBB=1, EAB=1

350

EAA=1, EBB=1, EAB=1.27

300 250 200 150 100

35

50

41

0 0

10

20

30

40

50

Size of maximum clusters

Figure 3. Size distributions of maximum molecular-uniform (squares) and

molecular-staggered (triangles) crystalline clusters under two enhancements of the corresponding driving forces as labeled when Ef/Ec = 0 and kT/Ec = 5.1. We obtained the data per 200 MCcs for 6×105 MCcs. For the details of definitions, see the text.

The favorite of intramolecular nucleation of molecular-uniform crystals can be revealed by the fraction of adjacent chain-folding on the stem-end surfaces of maximum crystalline clusters. To this end, we defined the adjacent chain-folding as two neighboring crystalline stem-ends connected by the small loops of 1 ~ 3 non-crystalline bonds. We chose two examples of large maximum crystalline clusters at the sizes 35 and 41, as indicated by the arrows in Figure 3. At the size 35, the mean fractions of adjacent chain-folding were 0.44 and 0.27 for maximum A-A and A-B clusters, respectively, while at the size 41, they become 0.58 and 0.39. Both results indicate more adjacent chain-folding in the maximum molecular-uniform crystalline 10


Page 11 of 27

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


clusters, confirming their favorites for intramolecular crystal nucleation during thermal fluctuations at high temperatures.

B. Crystal growth. Primary crystal nucleation involves only a few polymers, not

enough to represent major polymers performing chain-folding in crystallization. The more important process resulting in lamellar crystals is the subsequent crystal growth with its kinetics dominated by secondary crystal nucleation at the lateral growth front of lamellar crystals. Therefore, we set up a template to bypass primary crystal nucleation and to observe lamellar crystal growth induced by the template at a properly high temperature, as the snapshot of the sample system shown in Figure 4a. Enhancing EAA will result in mainly A-A molecular-uniform crystals grown from the template, while enhancing EAB will result in mainly A-B molecular-staggered crystals grown from the template, as the examples demonstrated separately in Figures 4(b) and 4(c). Apparently in the case of Figure 4(b), large-scale segregation of two components is not favorable for the growth kinetics of molecular-uniform crystals.

11



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

Page 12 of 27

12


Page 13 of 27

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


Figure 4. (a) Snapshot of the initial state of binary polymer blends in 646464 cubic

cells for our further observations of crystal growth, with an array of 8 folded A-chains spanning over Y axis at X=1 and Z=24. The red bonds were the template of 8 folded A-chains, blue ones were the other A-chains and yellow ones were for B-chains. (b) Snapshot of the lamellar crystals grown from the template under EAA=1.2, EBB=1 and EAB=1 for 3×105 MCcs when Ef/Ec = 0 and kT/Ec = 5.1, demonstrating a

molecular-uniform crystal containing mainly single components. Only those crystalline bonds containing more than 15 parallel neighbors irrespective A and B bonds are drawn. The color set follows (a). Viewing along Z axis. (c) Snapshot under EAA=1, EBB=1 and EAB=1.2, demonstrating a molecular-staggered crystal containing

mainly alternating components. The other conditions are the same as (b).

13



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

Page 14 of 27

Higher probabilities of intramolecular secondary nucleation will result in larger fractions of adjacent chain-folding. In order to confirm the preference of intramolecular secondary crystal nucleation at the growth front of molecular-uniform crystals, we calculated again the surface fractions of adjacent chain-folding on the single lamellar crystals grown from the template under separate enhancements of the two driving forces. As shown in Figure 5(a), we firstly set Ef/Ec=0.3 to diminish the influence of post-growth crystal thickening, because post-growth thickening will consume certain amount of adjacent chain-folding. One can see that the enhancement of EAA brings higher mean fractions of adjacent chain folding, in comparison to the parallel enhancement of EAB. We then set Ef/Ec=0, the difference between the mean fractions of adjacent chain-folding in two parallel cases remains right after the finish of crystal growth, although crystal thickening reduces it in a certain extent, as demonstrated in Figure 5(b). The results above confirm again that the molecular-uniform crystal growth generates more adjacent chain-folding due to their favorable intramolecular crystal nucleation.

14


Page 15 of 27

Mean fraction of adjacent chain-folding

a. 1.0

Prohibit crystal thickening 0.8

0.6

EAA EAB

0.4

0.2

0.0 1.10

1.15

1.20

1.25

1.30

Enhanced driving force / (Ep/Ec)

b.

Mean fraction of adjacent chain-folding

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


1.0

Allow crystal thickening 0.8

EAA EAB

0.6

0.4

0.2

0.0 1.10

1.15

1.20

1.25

Enhanced driving force /(Ep/Ec)

1.30

Figure 5. Mean fractions of adjacent chain folding on the stem-end surfaces of

molecular-uniform and staggered crystals grown under separate enhancement of the driving forces as labeled with other kinds of driving forces remained at 1 when (a) Ef/Ec = 0.3 and kT/Ec = 5.9 at the period 1×105 MCcs, and (b) Ef/Ec = 0 and kT/Ec = 5.1

at the period 1×105 MCcs. Each data was obtained from five individual simulations to 15



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

Page 16 of 27

make the error bar. The adjacent chain folding was judged by the small loops of 1 ~ 3 non-crystalline bonds connecting the neighboring crystalline stem-ends, while the crystalline bond contains more than 15 parallel neighbors irrespective of A or B bonds. The error bars are smaller than the size of symbols.

As a consequence of kinetic selection between two nucleation modes, the linear growth rates of template-induced lamellar crystal will be different between separate enhancements of two driving forces. We therefore traced the time evolution of crystal growth fronts, as the front defined by the largest distance away from the template for the minimum three consecutive crystalline bonds in parallel to the template. Since here the post-growth thickness does not change the favorite situation of chain-folding, our observations were kept going on with Ef/Ec=0. From Figure 6(a), one can see that the growth front advances with an almost linear dependence of time, reflecting a constant linear growth rate dominated by secondary crystal nucleation at the growth front. The slopes of those developing curves in Figure 6(a) are summarized in Figure 6(b). Indeed, as we expected, enhancing EAA brings higher crystal growth rates than parallel-enhancing EAB, implying the kinetic preference for intramolecular secondary crystal nucleation.

16


Page 17 of 27

Growth front /Lattice site

a.

35 30

EAA=1.10

25

E EAB=1.10

20 15 10 5 0 0

10

20

30

40

4


b. Linear growth rate /Lattice site/MCcs

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


0.0016

EAA 0.0012

EAB

0.0008

0.0004

0.0000 1.10

1.15

1.20

1.25

Enhanced driving force /(Ep/Ec)

1.30

Figure 6. (a) Time evolution curves of crystal growth fronts on X axis under separate

enhancement of two driving forces when others are kept at 1, Ef/Ec = 0 and kT/Ec = 5.1. Black lines are for 1.1 ~ 1.3 values under the only change of EAA and red ones are for 1.1 ~ 1.3 values of changing EAB only. (b) Linear crystal growth rates under separate enhancement of two driving forces as labeled. Each point was obtained from 17



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

Page 18 of 27

the slope of linear growth region, and was averaged over five individual simulations to get the error bar.

The linear growth rates of lamellar crystals could be determined by many factors according to the expression below.15

G  G0 (l  lmin )

F f exp( c ) (2) kT kT

where G0 is the prefactor containing the chain-length effect and the short-range diffusion barrier, l is the lamellar thickness, lmin is the minimum thickness, f is the fusion free energy per chain unit, Fc is the free energy barrier for intramolecular secondary crystal nucleation. In our present two cases, polymer chain lengths are uniform, and molecular-uniform crystal growth exhibits unfavorable long-distance diffusion for large-scale segregation, so G0 cannot be the reason for faster growth of molecular-uniform crystals. In parallel situations of half-half binary blends, separate enhancement of two driving forces benefit all crystalline bond pairs in the molecular-uniform crystals, and half pairs in the molecular-staggered crystals, so the thermodynamic factors like lmin and f will favor molecular-uniform crystals, but more chain folding still implies that intramolecular nucleation barrier Fc could be the main reason for faster growth of molecular-uniform crystals. In addition, the temperatures remain unchanged. The lamellar thickness l is a self-consistent result of crystal growth kinetics,23 which is the key factor to interpret the self-poisoning phenomenon of integer-folded short-chain crystal growth.24 We traced the time evolution of lamellar thicknesses of 18


Page 19 of 27

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


two crystals under parallel enhancements of driving forces, as the results shown in Figure 7(a). One can clearly see that enhancing EAB brings much higher capability of crystal thickening to the molecular-staggered crystals than parallel enhancing EAA for the molecular-uniform crystals. Figures 7(b) and 7(c) demonstrate the significant post-growth thickening of the single lamellar molecular-staggered crystal. New stems will be inserted into the thickened lamellar crystal to fill the vacancy raised by thickening, as observed in previous molecular simulations,24 which change the lateral arrangement of stem compositions as shown in the snapshots. Previous investigation of molecular simulations has proved that chain-folding will bring an additional free energy barrier to the lamellar thickening from once-folding to chain-extending.25 Thus, the high capability of lamellar thickening of molecular-staggered crystals can be attributed to their relatively smaller fractions of adjacent chain-folding in addition to their lower thermodynamic packing energy. Since molecular-uniform crystals are thinner than molecular-staggered crystals, the factor of lamellar thickness can be ruled out as well from the possible reasons for faster growth of molecular-uniform crystals. In summary of the considerations in the above two paragraphs, the free energy barrier Fc for intramolecular secondary nucleation could be the main reason for our present observations on faster growth of molecular-uniform crystals, as confirmed by their relatively larger fractions of adjacent chain-folding. This conclusion provides a direct molecular-level evidence to the kinetic preference of intramolecular crystal nucleation as the origin of dominant chain-folding for lamellar crystals.

19



a.

28

EAB=1.3 EAB=1.2

24

Mean stem length

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

Page 20 of 27

EAB=1.1 EAA=1.3

20

EAA=1.2 EAA=1.1

16 12 8 4 0

10

20

30

40

4


Figure 7. (a) Time evolution curves of mean stem length under separate enhancement

of two driving forces as labeled when others are kept at 1, Ef/Ec = 0 and kT/Ec = 5.1. The stem length was defined as the number of consecutive crystalline bonds in parallel with the template bonds. (b) Snapshot of single lamellar crystal grown under EAA=1, EBB=1 and EAB=1.2 when Ef/Ec = 0 and kT/Ec = 5.1 at the time period 1105 MCcs. Only the crystalline bonds containing more than 15 parallel neighbors irrespective of A and B bonds are drawn, with the red color for the template of 8th 20


Page 21 of 27

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


folded A-chains, the blue color for the other A-chains and the yellow color for all the B-chains. Viewing along Y axis. (c) Snapshot at the time period 4105 MCcs. The other conditions are the same as (b).

IV. CONCLUSIONS There are two competing nucleation modes for polymer primary and secondary crystal nucleation: intramolecular chain-folding and intermolecular fringed-micelle. The growth kinetics of polymer crystals will prefer to choose the former, which generates a large amount of adjacent chain-folding and results in the formation of characteristically lamellar crystals. By means of dynamic Monte Carlo simulations of symmetric binary polymer blends, we provided a direct molecular-level evidence for the kinetic preference of intramolecular nucleation, which is more favored by molecular-uniform crystals than by molecular-staggered crystals under parallel enhancements of driving forces. The results demonstrated faster nucleation and growth with more chain-folding for molecular-uniform crystals than for parallel-enhanced molecular-staggered crystals. We ruled out the other kinetic factors such as chain length, diffusion barrier, thermodynamic driving forces and lamellar thickness, and concluded that the lower free energy barrier for intramolecular secondary crystal nucleation is the sole reason for our present observations. Thus, by such a proper design in our simulations, we obtained a direct molecular-level evidence for the kinetic origin of dominant chain-folding, which results in the characteristic beta-sheet or lamellar crystals upon macromolecular crystallization in quiescent 21



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

Page 22 of 27

solutions and melt.

■ AUTHOR INFORMATION

Corresponding Author *E-mail: [email protected]. Notes The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The author WH thanks the stimulating discussion offered from Professor Go Matsuba at Yamagata University and Professor Pengju Pan at Zhejiang University. The financial support from National Natural Science Foundation of China (nos. 21274061 and 21474050), Program for Changjiang Scholars and Innovative Research Team, and Priority Academic Program Development of Jiangsu Higher Education Institutions, is appreciated.

■ REFERENCES 1

Wunderlich, B. Macromolecular Physics: Crystal Structure, Morphology, Defects;

Academic Press: New York; Vol. 1, 1973 2

Bassett, D. C. Principles of Polymer Morphology; Cambridge University Press:

Cambridge; 1981 3

Wunderlich, B. Macromolecular Physics: Crystal Nucleation, Growth, Annealing; 22


Page 23 of 27

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


Academic Press: New York; Vol. 2, 1976 4

Zachmann, H. G. Der Einfluss der Konfigurationsentropie auf das Kristallisations

und Schmelzmerhalten von Hochpolymeren Stoffen. Kolloid Z. Z. Polym. 1967, 216-217, 180-191 5

Hu, W.-B. Intramolecular Crystal Nucleation. In Lecture Notes in Physics:

Progress in Understanding of Polymer Crystallization; Strobl, G.; Reiter, G.; Eds.; Springer-Verlag: Berlin, 2007; Vol. 714, pp 47-63 6

Hu, W.-B.; Frenkel, D.; Mathot, V. B. F. Intramolecular Nucleation Model for

Polymer Crystallization. Macromolecules 2003, 36, 8178-8183 7

Lauritzen, J. I.; Hoffman, J. D. Theory of Formation of Polymer Crystals with

Folded Chains in Dilute Solution. J. Res. Natl. Bur. Stand., Sect. A 1960, 64, 73-102 8

Hoffman, J. D.; Lauritzen, J. I. Crystallization of Bulk Polymers with Chain

Folding: Theory of Growth of Lamellar Spherulites. J. Res. Natl. Bur. Stand., Sect. A 1961, 65, 297-336. 9

Hoffman, J. D.; Davis, J. T.; Lauritzen, J. I. The Rate of Crystallization of Linear

Polymers with Chain Folding. In Treatise on Solid State Chemistry; Hannay, N. B., Eds.; Plenum: New York, 1976; Vol. 3, pp 497-614 10

Hu, W.-B. Chain Folding in Polymer Melt Crystallization Studied by Dynamic

Monte Carlo Simulations. J. Chem. Phys. 2001, 115, 4395-4401 11

Hu, W.-B. Molecular Segregation in Polymer Melt Crystallization: Simulation

Evidence and Unified-scheme Interpretation. Macromolecules 2005, 38, 8712-8718 12

Cai, T.; Ma, Y.; Yin, P.-C.; Hu, W.-B. Understanding the Growth Rates of Polymer 23



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

Page 24 of 27

Co-crystallization in the Binary Mixtures of Different Chain Lengths. J. Phys. Chem. B 2008, 112, 7370-7376 13

Jiang, X.-M.; Li, T.-X.; Hu, W.-B. Understanding the Growth Rates of Polymer

Co-crystallization in the Binary Mixtures of Different Chain Lengths: Revisited. J. Phys. Chem. B 2015, 119, 9975–9981 14

Hu, W.-B.; Cai, T. Regime Transitions of Polymer Crystal Growth Rates:

Molecular Simulations and Interpretation Beyond Lauritzen-Hoffman Model. Macromolecules 2008, 41, 2049−2061 15

Ren, Y.-J.; Ma, A.-Q.; Li, J.; Jiang, X.-M.; Ma, Y.; Toda, A.; Hu, W.-B. Melting of

Polymer Single Crystals Studied by Dynamic Monte Carlo Simulations. Eur. Phys. J. E: Soft Matter Biol. Phys. 2010, 33, 189-202 16

Gao, H.-H.; Wang, J.; Schick, C.; Toda, A.; Zhou, D.-S.; Hu, W.-B. Combining

Fast-scan Chip-calorimeter with Molecular Simulations to Investigate Superheating Behaviors of Lamellar Polymer Crystals. Polymer 2014, 55, 4307-4312 17

Ren, Y.-J.; Zha, L.-Y.; Ma, Y.; Hong, B.-B.; Qiu, F.; Hu, W.-B. Polymer

Semi-crystalline Texture Made by Interplay of Crystal Growth. Polymer 2009, 50, 5871-5875 18

Nie, Y.-J.; Gao, H.-H.; Yu, M.-H.; Hu, Z.-M.; Reiter, G.; Hu, W.-B. Competition of

Crystal Nucleation to Fabricate the Oriented Semi-crystalline Polymers. Polymer 2013, 54, 3402-3407 19

Nie, Y.-J.; Gao, H.-H.; Hu, W.-B. Variable Trends of Chain-folding in Separate

Stages of Strain-induced Crystallization of Bulk Polymers. Polymer 2014, 55, 24


Page 25 of 27

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


1267-1272 20

Pan, P.; Han, L.; Bao, J.; Xie, Q.; Shan, G.; Bao, Y. Competitive

Stereocomplexation, Homocrystallization, and Polymorphic Crystalline Transition in Poly(l-lactic acid)/Poly(d-lactic acid) Racemic Blends: Molecular Weight Effects. J. Phys. Chem. B 2015, 119, 6462-6470 21

Hu, W.-B.; Frenkel, D.; Mathot, V. B. F. Sectorization of a Lamellar Polymer

Crystal Studied by Dynamic Monte Carlo Simulations. Macromolecules 2003, 36, 549-552 22

Hu, W.-B.; Frenkel, D. Polymer Crystallization Driven by Anisotropic Interactions.

Adv. Polym. Sci. 2005, 191, 1-35 23

Jiang, X.-M.; Reiter, G.; Hu, W.-B. How Chain-folding Crystal Growth

Determines the Thermodynamic Stability of Polymer Crystals. J. Phys. Chem. B 2016, 120, 566-571 24

Ma, Y.; Qi, B.; Ren, Y.-J.; Ungar, G.; Hobbs, J. K.; Hu, W.-B. Understanding

Self-poisoning Phenomenon in Crystal Growth of Short-chain Polymers. J. Phys. Chem. B 2009, 113, 13485-13490 25

Wang, M.-Q.; Gao, H.-H.; Zha, L.-Y.; Chen, E.-Q.; Hu, W.-B. Systematic Kinetic

Analysis on Monolayer Lamellar Crystal Thickening via Chain-sliding Diffusion of Polymers. Macromolecules 2013, 46, 164-171

25



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

Page 26 of 27

For Graphic Abstract:

26


Page 27 of 27

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


252x134mm (120 x 120 DPI)


Intramolecular Crystal Nucleation Favored by Polymer Crystallization

Recommend Documents