Fibril Crystal Growth in Diblock Copolymer Solutions Studied by

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Fibril Crystal Growth in Diblock Copolymer Solutions Studied by Dynamic Monte Carlo Simulations Rongfang Shu, Liyun Zha, Abdel Azeem Eman, and 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

ABSTRACT: Quasi-one-dimensional fibril crystal growth of diblock copolymers is a fundamental issue in the investigation of nanotechnology and neurodegenerative diseases. We performed dynamic Monte Carlo simulations of lattice polymers to study the crystallization-driven fibril crystal growth of diblock copolymers under two circumstances of solutions: sporadic crystals with the feeding mode of constant polymer concentrations, and massive crystals with the depleting mode of decaying polymer concentrations. We confirmed anisotropic driving forces as a prerequisite of steady growth of fibril crystals. The lamellar crystal width is confined by the noncrystalline block below a critical concentration that shifts down with the decrease of the noncrystallizable block fractions. In the depleting mode, the long-axis sizes at the early stage of fibril crystal growth can be fitted well by an exponential-decay function of time, and the growth rates decrease linearly with polymer concentrations by following the growth rates in the feeding mode, appearing as consistent with our previous simulation results of homopolymer solutions.

I. INTRODUCTION Diblock copolymers are able to self-assemble into nanoscale micelles with various geometrical features in solutions, which include spheres, cylinders,1 vesicles,2 and more complex shapes.3 These nanostructures have potential applications in drug delivery,4 toughening agents of epoxy resins,5 and hosts of metal nanoparticles.6 The micelle morphologies are determined by the block ratio, the solvent selectivity, the temperature and some other factors.7−11 Winnik and Manners et al. proved that the crystallization in one block could play a vital role in the above self-assembly and called it a crystallization-driven force for micelle formation.11−15 However, the formation of cylindrical micelles was found to be limited in a narrow range of length ratios of crystallizable to noncrystallizable blocks.16−19 Moreover, the sizes of the micellar core can be adjusted by the lengths of the crystallizable block.12 Zheng et al. suggested to reducing the length ratio of the noncrystalline block for a smaller curvature of interfaces, which results in transitions from spherical to cylindrical or even lamellar micelles.20 Mihut et al. observed various morphologies of spheres, rods and worms, by adjusting diblock ratios and the whole lengths of polybutadiene-b-poly(ethylene oxide).21 Yet, how the block ratio decides the cylinder formation upon crystallization is not so clear.1,12,22−24 © XXXX American Chemical Society

Beta-sheet crystalline strands of polypeptides constitute the amyloid fibers, which bring the primary influence on many neurodegenerative diseases, for instance Alzheimer’s disease.25 Crystal growth in the regular block of polypeptide is a key step for the therapeutic strategy of biological and chemical interventions.26 Therefore, fibril crystal growth of peptidebased block copolymer acts as a good model in understanding such a delicate case with the potential confinement effect of noncrystallizable blocks.27 Massive self-seeded cylindrical self-assembly driven by crystallization has been studied in the solutions of poly(isoprene-b-ferrocenyldimethylsilane) diblock copolymers.28,29 Inspired by these experiments, we have performed dynamic Monte Carlo simulations of homopolymer solutions to study the quasi-1D growth kinetics of lamellar crystals,30 and the results showed that the growth is mainly dominated by intramolecular secondary nucleation at the growth front. Here stepping straightforward, we employ the same approach to investigate quasi-1D crystal growth in diblock copolymer solutions. We will confirm the prerequisite condition of Received: March 6, 2015 Revised: April 14, 2015

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with the time unit defined as the total number of trial moves same as that of lattice sites. Polymer chains in the ordered state were first relaxed under athermal conditions over 106 MCcs to achieve random coils, except for an 1 × 64 × 8 layer of homopolymer chains spanning over Y-axis at X = 64 and Z = 28 in the 128 × 64 × 64 rectangular lattice for the crystal template. For clarity, only the −X side of the template charges the parallel-packing interactions to initiate crystal growth along this direction. The linear crystal growth rates were measured according to the time-evolution slope of the growth front, while the growth front was determined by −X location farthest from the template for the parallel-oriented stem holding at least three crystalline bonds (each contains more than 15 parallel neighbors). Two different circumstances of polymer solutions were considered for crystal growth in our simulations. The first circumstance mimics sporadic crystals with their growth maintaining an almost constant volume fraction of polymer chains in the solution phase. To this end, we randomly feed one chain into the space away from the crystal when a chain has joined into the crystal. This circumstance provides a constant thermodynamic condition for crystal growth. The second circumstance mimics massive crystal growth consuming polymers that hold a fixed total amount in solutions, so free polymer chains in the solution space were depleted with time evolution. This circumstance is more resemblance to the selfseeding experiments.28,29 In this sense, both the fiber-axis dimension and the width of lamellar crystals in the first circumstance intend to expand continuously with time evolution, while in the second case, they may stop at a certain concentration before all the crystalline blocks are completely exhausted. Since we set a low chain-mobility for postgrowth thickening, the crystal thickness remains constant as dominated mainly by the crystallization temperature.33 The template thickness is just large enough to initiate crystal growth and brings no influence to the crystal thickness.

anisotropic driving forces for the spontaneous and steady quasi1D growth of diblock copolymers, and prove the confinement effect of the noncrystallizable blocks on the lateral surfaces of lamellar crystals due to their overcrowding on the fold-end surfaces. The kinetics of fibril crystal growth in the depleting mode reproduces well our previous simulation results on homopolymer solutions. The rest of the paper is organized as follows: First, the computational details are briefly introduced and proper simulation parameters are determined. Then, the seeded cylindrical growth will be studied on three aspects: the driving forces, the confinement effects and the growth kinetics. By the end, we summarize our conclusions.

II. COMPUTATIONAL DETAILS The classical lattice model of polymer solutions was used in our Monte Carlo simulations. We put side-by-side 1200 (the amount subject to polymer volume fraction 0.293) once-folded polymer chains, each occupying consecutive 128 lattice sites with 112/16 ratio of noncrystallizable to crystallizable blocks (also subject to change), in a 128 × 64 × 64 lattice box. The rest vacant sites were tentatively regarded as athermal-mixing phantom solvent. Microrelaxation trial moves perform singlesite jumping or local sliding diffusion.31 Each bond connecting two consecutive chain units could orient along either 6 lattice axes, 12 face diagonals, or 8 body diagonals; therefore, the total coordination number of the lattice is 26, which is high enough to diminish the influence of discrete space on phase transitions. We rejected the trial moves causing hard-core overlaps or bond crossing. Periodic boundary conditions were employed in X-, Yand Z-axes of the lattice box. The conventional Metropolis sampling algorithm was employed for each microrelaxation step with the total potential energy change as given by Ep E ΔE = Δp + Δc c + kBT kBT kBT

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∑ f (i ) i=1

Ef kBT

III. RESULTS AND DISCUSSION A. Driving Forces. In polymer single crystals, crystal growth rates are often different between a-axis and b-axis. For instance, polyethylene is well-known for its growth along a-axis slowing down at high temperatures, which makes the lamellar crystals to develop along b-axis in alignment with the radius of spherulites. As a consequence of the anisotropic driving forces for crystal growth, the crystalline core of diblock copolymers initiates cylindrical micelles only along the long axis of polyferrocenyldimethylsilane single crystals.28 We first determined how much folds of the driving force along the long axis over the other dimensions as the prerequisite condition of the present fibril growth. In details, we stepwise adjusted the driving force parameter Ep/Ec on the long-axis −X direction from 1 to 5 under the feeding mode of polymer solutions, and meanwhile kept the driving force parameters on other axes as 1. We measured the linear crystal growth rates as the results shown in Figure 1. One can see that with the increase of Ep/Ec ratios over 3, spontaneous crystal growth along the −X direction occurs, and the growth rates increase with the enhancement of anisotropic driving forces. In addition, larger Ep/Ec intends to make more steady growth of fibril crystals, as demonstrated in the inset snapshots for the example of Ep/Ec = 5 in Figure 1. This observation confirms the strong enough anisotropy of driving forces as the prerequisite of fibril growth. Under the current conditions, the driving force

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E = (ΔpEp /Ec + Δc + ∑ f (i)Ef /Ec) c kBT i=1

(1)

Here, the energy parameter Ep describes the nonparallel packing energy for two neighboring bonds deviating from the crystalline state,32 and Δp is the net amount of nonparallel pairs of polymer bonds after motion; the energy parameter Ec characterizes the noncollinear connection energy for two consecutive bonds along the polymer chain, and Δc is the net amount of noncollinear connections after motion; Ef is the frictional barrier for sliding diffusion of the bond in the crystalline phase (supposing crystal thickening via chain-sliding diffusion), and Σf(i) is the sum of parallel neighbors of the bonds along the path of local sliding diffusion. We set Ef/Ec = 0.3 that is large enough to prohibit any postgrowth crystal thickening during the isothermal crystallization process. We selected the reduced temperature kBT/Ec = 3.3 (below simplified as T = 3.3) as the fixed system temperature to observe crystal growth driving by Ep/Ec = 1 under the variations of other parameters, where kB is Boltzmann’s constant. Since the mixing interactions among two blocks and the solvent were set as zero, there was no micelle formation prior to crystallization in the solutions. The effect of temperature variations as well as solvent selectivity will be investigated in another publication. Monte Carlo cycles (MCcs) were recorded B

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when the noncrystallizable blocks take the confinement effect on the fibril width, we fixed the temperatures and changed the length ratios of the noncrystallizable blocks as well as polymer concentrations. We switched the template size into 1 × 8 × 8 at X = 64, Y = 29, and Z = 28, in order to avoid the interactions of two or more fibril crystals upon parallel growth, as demonstrated in Figure 2. We defined the crystal width as the number of crystalline stems in the Y-axis, after averaged over the −X-axis from the template surface to the growth front. Each stem contained at least 3 consecutive crystalline bonds, while the crystalline bond held more than 15 parallel neighbors. The time evolution of crystal widths for a specific noncrystallizable composition at various polymer concentrations are summarized in Figure 3a. One can see that below a critical concentration (0.244), the crystal widths appear as saturated after 1 × 106 MCcs, implying a steady growth of cylinder crystals. Figure 3b summarized the crystal widths after the growth of 4.5 × 106 MCcs at various polymer concentrations for diblock copolymers containing different noncrystallizable block compositions as labeled. In the general impression, at each polymer concentration, longer noncrystallizable blocks result in smaller widths of crystals, reflecting an effect of confinement. For each noncrystallizable block composition, the crystal widths appear insensitive to polymer concentrations below a critical concentration, implying the complete confinement imposed by the noncrystallizable blocks. Above that critical value, the crystal widths exhibit almost linear dependence upon polymer concentrations, implying steady crystal growth without anymore confinement. Such a linear relationship reflects a linear concentration dependence of growth rates at the lateral growth fronts of the width, as observed for steady fibril growth of polymer single crystals in previous simulations.30 In other words, higher polymer concentrations provide stronger driving forces for crystal growth to overcome the shielding of the longer noncrystallizable blocks at the growth front of the width. The critical concentrations shift down with the decrease of noncrystallizable block compositions, showing weaker thermodynamic driving forces for crystal growth required to get over the weaker confinement effect of shorter noncrystallizable blocks. The critical concentration also implies a morphological transition of crystal growth from lamella to cylinder, due to the

Figure 1. Linear crystal growth rates of 128-mer diblock copolymers versus the folds of crystallization driving force (Ep/Ec) on the −X direction over the other axes, under the circumstance of feeding mode with polymer volume fraction 0.293 at T = 3.3. Each data point was averaged over three individual simulations. The upside inset snapshot demonstrates initial diblock copolymer solution with yellow folded chains as the crystal template. Crystallizable blocks containing 16 monomers are drawn in green and noncrystallizable blocks containing 112 monomers are drawn in blue. The downside inset snapshot shows only yellow crystalline bonds each containing more than 15 parallel neighbors at the time period 2 500 000 MCcs to exposure spontaneous and steady growth of fibril crystals under the driving force parameter 5 on −X, as indicated by the arrow.

parameter 5 provides a suitable growth rate along the long axis in the limited time window of our following observations. B. Confinement. In the real experiment, increasing the length of noncrystallizable block can suddenly switch micelle growth from lamella to cylinder, implying an overcrowding of the noncrystallizable blocks on the fold-end surfaces which shields the lateral surfaces and brings a confinement effect to the lateral advancing of the crystalline core.28 Since the driving force on that lateral front of the width is weaker than that of the fiber axis, the shielding will stop the width growth in priority. The overcrowding depends first upon the length fractions of the noncrystallizable blocks and second upon the thermodynamic conditions (temperature and concentration). To find out

Figure 2. Snapshots of spontaneous fibril crystal growth crystallized at T = 3.3 for 2100000 MCcs, from the initial polymer solution of the volume fraction 0.293 under the feeding mode in a 128 × 64 × 64 rectangular lattice. Ep/Ec = 5 on the −X direction and Ep/Ec = 1 on the other axes. The chain length is 128, holding noncrystallizable block 112 monomers. The yellow crystal is seeded by the size 1 × 8 × 8 at X = 64, Y = 29, Z = 28, and the crystalline stems are in alignment with the Z-axis. (a) Snapshot showing only those crystalline bonds holding more than 15 parallel neighbors. (b) Snapshot showing all those bonds joining the fibril crystal. The crystalline part of crystallizable blocks is drawn in yellow, and the rest are drawn in green. Noncrystalline blocks are drawn in blue. C

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confinement effect of the noncrystallizable blocks. In this sense, the fibril crystal growth (0.293 for 112/128) shown in Figure 2 actually belongs to the case above the critical concentration, so it still contains the potentiality to become a lamellar crystal, given by a long enough growth period. In the depleting mode, the spontaneous and steady growth of fibril crystal still occurs as demonstrated for example shown in Figure 4. In the depleting mode, the crystal widths vary with the decaying of polymer concentration. Therefore, we observed only the averaged widths of the whole crystal with time evolution. Again, the time evolution of crystal widths shows a critical initial polymer concentration (around 0.32) for the saturation of crystal widths, as shown in Figure 5a. The crystal widths obtained after the same periods of crystal growth at various polymer concentrations for different compositions of noncrystallizable blocks are summarized in Figure 5b. The critical concentrations again decrease with the decrease of noncrystallizable block compositions. One can see that the fibril crystal growth (0.293 for 112/128) shown in Figure 4 now belongs to the case below the critical concentration, and its lateral size is indeed confined by the noncrystallizable blocks. Figure 6 compares the critical concentrations between the feeding mode and the depleting mode for diblock copolymers containing various noncrystallizable block fractions. One can see that two modes are almost parallel to each other, and the depleting mode requires higher initial polymer concentrations to overcome the confinement effects. The results are reasonable, as crystal growth in the depleting mode consumes polymers and makes lower effective polymer concentrations. C. Growth Kinetics. In order to better understand the growth mechanism besides the confinement effect of noncrystalline blocks in real experiments, we further studied the variation of the growth rates in the depleting mode and compared it to the parallel results in the feeding mode. Since only the crystallizable blocks in the solutions play the thermodynamic role in the crystal growth kinetics, we counted the volume fraction of crystallizable blocks in the solutions for the following study, for a comparison of the present results to our previous work.30 Figure 7 demonstrates a typical timeevolution curve of the long-axis crystal sizes with the initial crystallizable-block concentration 0.0183 at T = 3.3 in the

Figure 3. (a) Time evolution curves of the crystal widths in the feeding mode at various volume fractions of polymers with the chain length 128 lattice sites (including the noncrystalline block 112 lattice sites) as labeled and T = 3.3. Ep/Ec on the −X-axis is 5 and Ep/Ec on other dimensions are 1. (b) Crystal widths observed at 4500000 MCcs for crystal growth under various volume fractions of polymers and various length fractions of noncrystalline blocks in a feeding mode for the chain length 128 lattice sites and T = 3.3. Each point was averaged over 3 times of individual simulations to make an error bar. The solid lines are drawn to guide the eyes.

Figure 4. Snapshots of spontaneous fibril crystal growth crystallized at T = 3.3 for 1590000 MCcs, from the initial polymer solution of the volume fraction 0.293 under the depleting mode in a 128 × 64 × 64 rectangular lattice. Ep/Ec = 5 on −X direction and Ep/Ec = 1 on the other directions. The chain length is 128, holding noncrystallizable block 112 monomers. The yellow crystal seed with the size 1 × 8 × 8 at X = 64, Y = 29, Z = 28 and the crystalline stems are aligned along the Z-axis. (a) Snapshot showing only those crystalline bonds holding 15 parallel neighbors. (b) Snapshot showing all those bonds joining the fibril crystal. The crystalline part of crystallizable blocks is drawn in yellow, and the rest are drawn in green. Noncrystalline blocks are drawn in blue. D

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Figure 7. Time-evolution curve of the long-axis crystal growth size with the initial volume fraction 0.0183 of the crystallizable blocks for the chain length 128 containing the crystallizable block 16 at T = 3.3. The insert is an enlargement of the early stage (the red box) fitted well by a single-exponential curve (the red line) with the equation L = 22.1(1 − e−0.33t) under the adjusted R2 value of 0.9436.

depleting mode. The crystal growth size is saturated when copolymers are depleted in the solution. Here, the copolymer volume fraction is 0.1484 below the critical concentration (around 0.33 for 112/128 in Figure 5b) of the confinement, so the width of crystal is nearly constant for quasi-1D crystal growth. The curve at the early stage in Figure 7 can be fitted well by a single-exponential decay function of time, as shown in the inset figure, implying a simple first-order kinetics as derived in our previous work.30 The first-order kinetics above is derived on the basis of the linear concentration-dependence of the growth rates, as observed in the low concentration region of the feeding mode shown in Figure 8. In the depleting mode, the linear crystal growth rates were obtained from the first derivative of the long-axis growth size to the time. In Figure 8, one can see

Figure 5. (a) Time evolution curves of the crystal widths in the depleting mode at various initial volume fractions of polymers with the chain length 128 lattice sites (including the noncrystalline block 112 lattice sites) as labeled and T = 3.3. Ep/Ec on the −X-axis is 5 and Ep/ Ec on other dimensions are 1. (b) Crystal widths observed at 4.5 × 106 MCcs for crystal growth under various volume fractions of polymers and various length fractions of noncrystalline blocks in a feeding mode for the chain length 128 lattice sites and T = 3.3. Each point was averaged over 3 times of individual simulations to make an error bar. The solid lines are drawn to guide the eyes.

Figure 8. Comparison of the linear crystal growth rates versus the volume fraction of the crystallizable blocks in solutions between the feeding mode and depleting mode for the copolymer length 128 containing the crystallizable block 16 at T = 3.3. The black squares were obtained in the feeding mode, and the blue curve was obtained in the early stage of the depleting mode. The arrow indicates the decay trend in the depleting mode. The straight line is drawn to guide the eyes.

Figure 6. Comparison of the critical volume fractions of diblock copolymers at different ratios of noncrystalline blocks between two modes.

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The Journal of Physical Chemistry B that the early stage growth rates of the fibril crystal in the depleting mode follow closely to the growth rates in the feeding mode showing the linear relationship to crystallizable block concentrations, implying the thermodynamic condition dominating the decaying of growth rates at that early stage in the depleting mode. The linear concentration-dependent behavior is consistent with quasi-1D crystal growth of homopolymer solutions in our previous simulations, indicating again the intramolecular-nucleation-controlled mechanism of crystal growth at the early stage of depleting modes as well as at low polymer concentrations of the feeding mode.

(4) Geng, Y.; Dalhaimer, P.; Cai, S. S.; Tsai, R.; Tewari, M.; Minko, T.; Discher, D. E. Shape Effects of Filaments Versus Spherical Particles in Flow and Drug Delivery. Nat. Nanotechnol. 2007, 2, 249−255. (5) Thio, Y. S.; Wu, J. X.; Bates, F. S. Epoxy Toughening Using Low Molecular Weight Poly(hexylene oxide)-Poly(ethylene oxide) Diblock Copolymers. Macromolecules 2006, 39, 7187−7189. (6) Cao, L.; Massey, J. A.; Winnik, M. A.; Manners, I.; Riethmuller, S.; Banhart, F.; Spatz, J. P.; M?ller, M. Reactive Ion Etching of Cylindrical Polyferrocenylsilane Block Copolymer Micelles: Fabrication of Ceramic Nanolines on Semiconducting Substrates. Adv. Funct. Mater. 2003, 13, 271−276. (7) Zhang, L. F.; Eisenberg, A. Multiple Morphologies of Crew-Cut Aggregates of Polystyrene-b-poly(acrylic acid) Block Copolymers. Science 1995, 268, 1728−1731. (8) Zhang, L. F.; Yu, K.; Eisenberg, A. Ion-Induced Morphological Changes in ’’Crew-Cut’’ Aggregates of Amphiphilic Block Copolymers. Science 1996, 272, 1777−1779. (9) Massey, J. A.; Temple, K.; Cao, L.; Rharbi, Y.; Raez, J.; Winnik, M. A.; Manners, I. Self-Assembly of Organometallic Block Copolymers: The Role of Crystallinity of the Core-Forming Polyferrocene Block in the Micellar Morphologies Formed by Poly(ferrocenylsilane-b-dimethylsiloxane) in N-Alkane Solvents. J. Am. Chem. Soc. 2000, 122, 11577−11584. (10) Abbas, S.; Li, Z. B.; Hassan, H.; Lodge, T. P. Thermoreversible Morphology Transitions of Poly(styrene-b-dimethylsiloxane) Diblock Copolymer Micelles in Dilute Solution. Macromolecules 2007, 40, 4048−4052. (11) Yuan, J. Y.; Xu, Y. Y.; Walther, A.; Bolisetty, S.; Schumacher, M.; Schmalz, H.; Ballauff, M.; Muller, A. H. E. Water-Soluble Organo-Silica Hybrid Nanowires. Nat. Mater. 2008, 7, 718−722. (12) Cao, L.; Manners, I.; Winnik, M. A. Influence of the Interplay of Crystallization and Chain Stretching on Micellar Morphologies: Solution Self-Assembly of Coil-Crystalline Poly(isoprene-block-ferrocenylsilane). Macromolecules 2002, 35, 8258−8260. (13) Raez, J.; Manners, I.; Winnik, M. A. Nanotubes from the SelfAssembly of Asymmetric Crystalline-Coil Poly(ferrocenylsilanesiloxane) Block Copolymers. J. Am. Chem. Soc. 2002, 124, 10381− 10395. (14) Qi, F.; Guerin, G.; Cambridge, G.; Xu, W.; Manners, I.; Winnik, M. A. Influence of Solvent Polarity on the Self-Assembly of the Crystalline−Coil Diblock Copolymer Polyferrocenylsilane-b-polyisoprene. Macromolecules 2011, 44, 6136−6144. (15) Hsiao, M. S.; Yusoff, S. F. M.; Winnik, M. A.; Manners, I. Crystallization-Driven Self-Assembly of Block Copolymers with a Short Crystallizable Core-Forming Segment: Controlling Micelle Morphology through the Influence of Molar Mass and Solvent Selectivity. Macromolecules 2014, 47, 2361−2372. (16) Choucair, A.; Eisenberg, A. Interfacial Solubilization of Model Amphiphilic Molecules in Block Copolymer Micelles. J. Am. Chem. Soc. 2003, 125, 11993−12000. (17) Jain, S.; Bates, F. S. Consequences of Nonergodicity in Aqueous Binary Peo-Pb Micellar Dispersions. Macromolecules 2004, 37, 1511− 1523. (18) Zhulina, E. B.; Adam, M.; LaRue, I.; Sheiko, S. S.; Rubinstein, M. Diblock Copolymer Micelles in a Dilute Solution. Macromolecules 2005, 38, 5330−5351. (19) Qian, J. S.; Zhang, M.; Manners, I.; Winnik, M. A. Nanofiber Micelles from the Self-Assembly of Block Copolymers. Trends Biotechnol. 2010, 28, 84−92. (20) Zheng, Y.; Won, Y. Y.; Bates, F. S.; Davis, H. T.; Scriven, L. E.; Talmon, Y. Directly Resolved Core-Corona Structure of Block Copolymer Micelles by Cryo-Transmission Electron Microscopy. J. Phys. Chem. B 1999, 103, 10331−10334. (21) Mihut, A. M.; Crassous, J. J.; Schmalz, H.; Drechsler, M.; Ballauff, M. Self-Assembly of Crystalline-Coil Diblock Copolymers in Solution: Experimental Phase Map. Soft Matter 2012, 8, 3163−3173. (22) Cameron, N. S.; Corbierre, M. K.; Eisenberg, A. Asymmetric Amphiphilic Block Copolymers in Solution: A Morphological Wonderland. Can. J. Chem.Rev. Can. Chim. 1999, 77, 1311−1326.

IV. CONCLUSIONS By means of dynamic Monte Carlo simulations, we investigated the anisotropic driving forces, the confinement effects and the kinetics on the quasi-1D fibril crystal growth of diblock copolymers in solutions with both feeding and depleting modes. First, a strong enough anisotropy in the driving forces of crystallization appears as the prerequisite of fibril crystal growth in the diblock copolymer solutions. On the basis of this fact, we successfully raised the spontaneous and steady growth of crystalline cores of cylindrical micelles from the template. Second, we identified the critical conditions for the confinement effect of noncrystallizable blocks on the crystal growth along the width dimension. Finally, with the confinement above, the growth kinetics of long-axis crystal sizes were compared between the feeding and the depleting modes, and the results are consistent with our previous observations on quasi-1D growth of lamellar crystals in homopolymer solutions. On the basis of the present work, we can go further to investigate the effect of solvent selectivity as well as the competition between two diblock copolymers containing different noncrystalline blocks upon fibril growth, to facilitate our better understanding on the fundamental aspects of the related experimental observations.

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AUTHOR INFORMATION

Corresponding Author

*(W.H.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the helpful discussion with Prof. Jieshu Qian at Nanjing University of Aeronautics and Astronautics, and Dr. Ran Ni at Amsterdam University. The financial support from National Natural Science Foundation of China (No. 21274061 and 21474050), National Basic Research Program of China (No. 2011CB606100), Program for Changjiang Scholars and Innovative Research Team, and Priority Academic Program Development of Jiangsu Higher Education Institutions is appreciated.

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