Systematic Kinetic Analysis on Monolayer Lamellar Crystal Thickening

Dec 18, 2012 - ... Physics of Ministry of Education, College of Chemistry, Peking University, Beijing 100871, China. Macromolecules , 2013, 46 (1), pp...
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Systematic Kinetic Analysis on Monolayer Lamellar Crystal Thickening via Chain-Sliding Diffusion of Polymers Mingqiu Wang,†,‡ Huanhuan Gao,† Liyun Zha,‡ Er-Qiang Chen,§ and Wenbing Hu*,† †

Department of Polymer Science and Engineering, State Key Lab of Coordination Chemistry, School of Chemistry and Chemical Engineering, Nanjing University, 210093 Nanjing, China ‡ Department of Chemical and Pharmaceutical Engineering, Chengxian College, Southeast University, 210088 Nanjing, China § Key Laboratory of Polymer Chemistry and Physics of Ministry of Education, College of Chemistry, Peking University, Beijing 100871, China S Supporting Information *

ABSTRACT: Lamellar polymer crystals are metastable due to their limited lamellar thickness. We performed dynamic Monte Carlo simulations of lattice linear polymers to investigate the kinetics of isothermal thickening via chainsliding diffusion in single lamellar crystals of polyethylene and poly(ethylene oxide). We sorted out three typical cases for controversial experimental observations. The basic case is a continuous increase of lamellar thickness for heavily folded long chains, with a logarithmic time dependence typical at the lateral growth front. Its kinetics is dominated by the activation energy barrier for sliding diffusion with higher speeds at higher temperatures. For integerfolded short chains, however, the lamellar thickness increases discontinuously, and its kinetics is dominated by a free energy barrier for surface nucleation. The latter can be further split into two cases: the thickening in the melt is mainly driven by the bulk free energy, with lower speeds at higher temperatures due to a temperature-sensitive barrier; while the thickening on a solid substrate is mainly driven by the surface free energy, with higher speeds at higher temperatures due to a temperature-insensitive barrier. The simulations facilitate our systematic understanding to the case-by-case microscopic mechanisms for the thickening of monolayer lamellar crystals via sliding diffusion of polymers. of monolayer lamellar crystals via chain-sliding diffusion.6 However, if not immediately fed with other amorphous (part of) chains, such a mechanism results in a local stem vacancy, which increases surface exposure of the crystal and thus be unfavorable for the stability of the system. For parallel-stacked multilayer crystals, a cooperative extending within the nearestneighbor lamellar crystals can avoid such a stem vacancy.7 For monolayer crystals, the stem vacancy can either be filled by the nearby amorphous chains in the melt to harvest more crystallinity8 or be accumulated into the inner holes and rough edges on a solid substrate after prepared in dilute solutions or thin films.9−15 So far, the thickening kinetics of this kind of polymers is still unclear, as its mechanism is controlled by many factors, such as temperature, concentration, pressure, chain length, chain mobility in the crystal, and substrate interactions. Even for monolayer single lamellar crystals, experimental observations on the temperature dependence of lamellae thickening rates appear controversial among various cases. The basic case is the continuous thickening of heavily folded long chains, which is controlled by an activation energy barrier

I. INTRODUCTION Crystallization of flexible linear polymers from quiescent melt and solutions prefers to generate lamellar crystals with their thermal stability mainly determined by a limited lamellar thickness.1 Upon isothermal annealing, the metastable lamellar crystals intend to thicken into more stable states with larger lamellar thickness, either via chain-sliding diffusion in the crystal or via melting−recrystallization process.2 The thickening mechanism is not only a fundamental issue to understand polymer crystallization but also a key procedure to control the semicrystalline texture as well as its performance for polymeric materials. The chain-sliding diffusion in the crystals is very much related to a high mobility of several well-investigated polymers such as polyethylene (PE) and poly(ethylene oxide) (PEO). Early in 1960s, by means of small-angle X-ray scattering (SAXS), Fischer and Schmidt observed faster lamellar thickening at higher temperatures upon annealing PE crystals.3 The lamellar thickness increases continuously and exhibits a logarithmic time dependence.3 Peterlin suggested that the temperature and time dependence can be understood as controlled by an activation energy barrier for chain diffusion in monolayer crystals.4 A more general diffusion barrier theory was developed by Sanchez, Colson, and Eby.5 Dreyfus and Keller even proposed a microscopic mechanism for thickening © 2012 American Chemical Society

Received: September 13, 2012 Revised: November 30, 2012 Published: December 18, 2012 164

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substrate, but faster at higher temperatures. It is obvious that three cases above contain different mechanisms, even though they all perform crystal thickening via chain-sliding diffusion. Therefore, the case-by-case thickening mechanisms in the experimental observations are worthy of further investigations. The advances in molecular simulations of polymer crystallization allow a direct visualization of lamellar crystal thickening at the molecular scale and thus facilitate our better understanding to the case-by-case microscopic mechanisms. We adopted dynamic Monte Carlo simulations of lattice linear polymers to reproduce various thickening kinetics in three cases separately corresponding to the lateral growth front of longchain lamellar crystals, LMW integer-folded-chain crystals grown in the melt, and LMW integer-folded-chain crystals grown in ultrathin films. The long-chain thickening at the lateral growth front reproduces the basic kinetics of a continuous thickening with an activation energy barrier for chain-sliding diffusion in the crystal. Both short-chain thickening exhibit a nucleation-controlled mechanism, but the driving force for thickening in the melt is the bulk free energy, while the driving force for thickening on a solid substrate is the surface free energy. This driving-force difference results in the nucleation barriers with different sensitivities to the temperature, leading to the opposite temperature dependences of the thickening nucleation rates. The paper is organized as follows. After the Introduction, we make a brief description on the simulation techniques, followed by the simulation results for three separate cases. The paper ends with a summary of our conclusions.

for chain-sliding diffusion in the monolayer lamellar crystals. By means of electron micrograph, Wunderlich and co-workers observed a wedge-shaped profile at the lateral growth front of monolayer thickened lamellar PE crystals grown under high pressure.16 Molecular simulations have reproduced such a continuous thickening, beginning at the lateral growth front and ending at the center of monolayer chain-folded crystals, which results in a harvest of self-seeding nuclei in the vicinity of the crystal center upon melting at high temperatures.17 The profile of the crystal growth front will reflect the time evolution of lamellar thickness. The thickening kinetics is dominated by a rate-determining step that is the slowest step along the path of crystal thickening. The specific metastability of folding lengths at integer fractions of chain lengths may change the rate-determining step in the basic case of crystal thickening. In contrast to the basic case above, thickening of low molecular weight (LMW) PEO from once-folded-chain to extended-chain lamellae appears discontinuous. Kovacs and co-workers observed the incubation period for the initiation of such crystal thickening in the melt, implying a nucleation-controlled process; in addition, they found that a lateral spreading rate of the thickened crystal is constant over time but gets reduced with the increase of temperature.18,19 The discontinuous thickening of metastable crystals brings significant changes in both morphology and linear crystal growth rates.20−23 Hikosaka and co-workers found that under high pressure the thickening growth rate of extended-chain PE crystals increases at lower temperatures (the increase of ∞ supercooling ΔT = T∞ m − T, where Tm represents the equilibrium melting point), similar to the conventional crystal growth rates dominated by the secondary crystal nucleation at the lateral growth front.24,25 Recent development of time-resolved experiments enables more direct kinetic observation on the above discontinuous thickening of single lamella on a solid substrate after prepared in dilute solutions or thin films. Wang and co-workers measured the average thickness of LMW PEO monolayer crystals in thin films by quasi-time-resolved SAXS using a synchrotron radiation source.26 They found the lamellar thickness doubled from once-folded to extended short chains, with the thickening rates (I) decreasing at lower temperatures (opposite to the melt case above!) and obeying the Arrheniustype (ln I ∝ T−1) temperature dependence. They proposed the thickening kinetics as dominated by long-distance mass diffusion in the process of melting−recrystallization for crystal thickening at the sites different from the melted crystals.26 Chen and co-workers traced the on-site thickening of LMW PEO monolayer crystals from once-folded to extended chains on the mica surface using atomic force microscopy (AFM).27 They directly observed the formation of distinctively thickened domains in the mother lamella, without long-distance mass diffusion, consistent with previous AFM observations.11−15 The thickening process is controlled by a nucleation-controlled mechanism via on-site sliding diffusion in the crystal, and the nucleation rates exhibit an Arrhenius-type temperature dependence as well.27 The controversial experimental observations above can be classified into three typical cases: a continuous thickening at the lateral growth front of long-chain lamellar crystals, faster at higher temperatures; a discontinuous thickening of monolayer folded-to-extended short-chain lamellar crystals in the melt, slower at higher temperatures; and the similar discontinuous thickening of monolayer short-chain crystals on a solid

II. SIMULATION TECHNIQUES Dynamic Monte Carlo simulations of the lattice polymer model have been widely applied in the study of polymer crystallization.28 In the lattice space, N consecutive occupations of the lattice sites represent each polymer chain containing N monomers, while the single vacant sites represent the free volume necessary for chain motion in the bulk phase. The motion of polymer chains follows a microrelaxation model29 involving both single-site jumping and local slithering to imitate the crankshaft rotation and chain-sliding diffusion of local polymer chains, respectively. Double occupation and bond crossing are forbidden in the cubic lattice box with periodic boundary conditions unless specified case by case. We employed the conventional Metropolis sampling algorithm to decide the acceptance probability of each microrelaxation step according to the minimum between one and exp(−ΔE/(kbT)). The potential energy barrier was formed by ΔE = (cEc + pEp + k bT ⎛ Ep = ⎜⎜c + p + Ec ⎝

n

∑ f (i )E f

n

∑ f (i ) i=1

+ b1B1 + b2B2 )/k bT

i=1

Ef Ec

+ b1

B1 B ⎞ E + b2 2 ⎟⎟ c Ec Ec ⎠ k bT (1)

where Ec is the noncollinear connection energy for a pair of bonds consecutively connected along the chain (reflecting the static flexibility of polymers), and c is the change of corresponding bond pairs; Ep is parallel attractions between polymer bonds (reflecting the driving force for polymer crystallization), p is the change of such bond pairs, and Ef is the frictional barrier for c-slip diffusion of the chains in the 165

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crystalline phase (reflecting the ability of crystal thickening via chain-sliding diffusion); ∑f(i) is the sum of parallel neighbors along the path of local sliding diffusion (for chain sliding along the stem in the crystal, the total barrier is proportional to the stem length); parameters B1 and B2 are used only in the third simulation case for thin films: B1 represents the repulsive energy between monomers and voids (referring to the air above the freestanding surface of the film), B2 represents the attractive energy between monomers and substrate sites (referring to the substrate beneath the film to avoid dewetting), and b1 and b2 are the changes of corresponding contact pairs. Reduced temperature kbT/Ec was used to adjust the system temperature (we omit its units below), where kb is Boltzmann’s constant and T the temperature. Monte Carlo cycles (MCc) were used to trace time evolution in our simulations, each of which was defined as the number of trial moves equal to the total number of lattice sites in the sample systems. Moreover, we defined a crystalline bond if it was packed with more than 15 parallel neighbors. Note that each bond may have a maximum of 24 parallel neighbors in the cubic lattice. The crystallinity of the system can then be calculated as the fraction of crystalline bonds in the total bonds. In the following, we report simulation results in three separate cases corresponding to the typical experimental cases above with variable thickening kinetics.

III. SIMULATION RESULTS A. Continuous Thickening of Long-Chain Monolayer Lamellar Crystals. For mobile high molecular weight (HMW) polymers in the crystal, lamellar thickening at the lateral growth front appears as the common case for continuous thickening via chain-sliding diffusion, resulting in a wedge-shaped profile of the growth front. In order to visualize such a thickening process at the growth front of a monolayer lamellar crystal, we put 1920 polymer chains (chain length 128) into a 643 cubic lattice box with periodic boundary conditions. The occupation density was as high as 0.9375 to mimic the bulk phase. By a long-term athermal relaxation, the chains were relaxed from the preset ordered state to a random-coil state. We set Ep/Ec = 1 to enable flexible chains as well as Ef/Ec = 0.02 to allow chain-sliding diffusion with a relatively low activation energy barrier. Crystallization was induced by a template that was formed by a layer of regularly folded chains with stem length 16 lattice sites, separately at temperatures of 4.6, 4.8, and 5.0. In the present observations, the chains were long enough (128 monomers) to perform multiple chain folds (up to seven) with limited fold lengths (around 16 lattice sites). The profile of the lamellar growth front is mainly dominated by instant crystal thickening via chain-sliding diffusion in the crystal, right after secondary crystal nucleation at the tip top. Since the advancing of the growth front keeps a constant speed over the time period, the lamellar thickness profile at the growth front reflects the crystal thickening at the early stage of growth. Figure 1a demonstrates a wedge-shaped profile at the lateral lamellar growth front. Figure 1b compares the different growth-front profiles obtained at three temperatures, which can be fitted into a logarithmic function of distances to the growth front quite well. Such a function implies a logarithmic time dependence of crystal thickness. The prefactors of the fitting logarithmic functions indicate that the thickening rates increase with temperature. This continuous thickening follows a common sliding-diffusion-controlled mechanism, and an

Figure 1. (a) Snapshot of the wedge-shaped growth profiles of relatively HMW polymer lamellar crystals (chain length 128 for heavy chain folding) grown at 35 000 MCc and T = 4.6 Ec/kb (Ep/Ec = 1, Ef/ Ec = 0.02) in the cubic lattice of 643. The template was placed at the left-end plane. Only the crystalline bonds are drawn in yellow cylinders. (b) Logarithmic function profiles of lamellar growth fronts of monolayer crystals obtained at T = 4.6, 4.8, and 5.0 Ec/kb, respectively.

activation energy barrier for sliding diffusion dominates its kinetics, consistent with the experiments and the related theoretical prediction.3−5 The logarithmic time dependence can be derived from a frictional barrier (ΔEs) on chain-sliding diffusion for lamellar thickening, which is presumably proportional to the lamellar thickness (l) set with ∑f(i) in eq 1 for our simulations:

ΔEs ∝ l

(2)

The linear thickness dependence of the frictional barrier has been adopted in Hikosaka’s theoretical consideration as well.24,25 The thickening rate (R) at a given temperature can be expressed as an activation process4,5 R=

dl ∝ e−ΔEs / kbT dt

(3)

Therefore, from the substitution between two equations above dl = be−al / kbT dt

(4)

one can derive the solution 166

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l = c ln t + d

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(5)

where a, b, c, and d are the proportional coefficients. A higher temperature enables polymer chains to move more actively, resulting in a positive T-dependence of the thickening rate. B. Discontinuous Thickening of Monolayer IntegerFolded Short-Chain Crystals in the Melt. Although crystal thickening is initiated via chain-sliding diffusion in the crystal, the thickening behaviors are also restricted by other factors, which could be responsible for the change of the ratedetermining step. One typical factor is the specific metastability with the folding lengths at the integer fractions of short chain lengths. Such a preference of fold lengths makes the thickening via chain-sliding diffusion appear discontinuous. The new lateral surface generated by discontinuous crystal thickening brings about a free energy barrier as in a nucleation-controlled mechanism. Meanwhile in the melt, the thickening of lamellar crystals harvests crystallinity, and the gain in the bulk free energy can get over the penalty of the new lateral surface free energy, like the conventional primary nucleation for crystal growth. However, this thickening is initiated via surface nucleation on the basal face of folded-chain lamellar crystals, which is different from the subsequent spreading rates of the thickened crystal dominated by the conventional surface nucleation at its lateral thickened front. To visualize the above-mentioned thickening case, a total number of 1920 chains, each containing 16 monomers to model LMW polymers, were placed into the 323 cubic lattice box with periodic boundary conditions. The occupation density is 0.9375 to mimic the bulk polymer phase. The preset ordered chains were first relaxed into random coils under athermal conditions, except for a layer of once-folded-chain template with a height of 8 lattice sites (see Figure 2a). At the reduced temperature of 4.50 with Ep/Ec = 1 and Ef/Ec = 0.02, the template induced the initially once-folded lamellar crystal growth (Figure 2b). The template chains became mobile together with other chains upon further isothermal annealing at an elevated temperature 4.58. Upon isothermal thickening, a thickened extended-chain domain emerges on the surface of the once-folded-chain lamellar crystal after a certain incubation period (Figure 2c), followed with its continuous development over the whole crystal (Figure 2d). In principle, the downside has the same probability to generate the thickened domain, although its nucleation is a rare event as well. In our present simulations of such a small system, we observed that the thickening occurs only at one side. We also provide the corresponding movie for the discontinuous crystal thickening as Supporting Information. Figure 3a shows an incubation period required to initiate a raise of crystallinity upon crystal thickening from once-foldedchain crystal. The incubation period implies a nucleationcontrolled mechanism. The nucleation rate (I) can be estimated from the average incubation periods over more than 30 independent observations. We defined the incubation period (ti) according to the onset of crystallinity rising in the time-evolution curve of crystallinity (see Figure 3a). So the average nucleation rate −1

⟨I ⟩ = ⟨ti⟩

Figure 2. Snapshots of the lamellar crystal growth and subsequent annealing of LMW polymers (chain length 16 monomers for initially once folding) in the melt phase. All the bonds were drawn in cylinders, with the amorphous part in blue and the crystalline part in yellow. (a) The initial melt of LMW polymers in a 323 cubic box with a template for crystal growth. The template is formed by once-folded chains in yellow. (b) The once-folded-chain lamellar crystal grown for a time period of 19 000 MCc at T = 4.50 (Ep/Ec = 1, Ef/Ec = 0.02). (c) The extended-chain thickening domain on the top surface of lamellae after annealing over 140 000 MCc at T = 4.58. For clarity, only the crystalline part is shown. (d) The extended-chain lamellar crystal completes its thickening after annealing over 804 000 MCc at T = 4.58.

as a function of supercooling ΔT = Tm(∞) − T, where Tm(∞) = 5.7 (used in the same approach in our previous simulations30), as shown in Figure 3c. The plotting shows that the logarithmic nucleation rate ln(I) reduces linearly with the increase of 1/T/ΔT. That is to say, the nucleation rate in the melt phase exhibits a conventional secondary-nucleationtype temperature dependence, as further discussed below. Note that there is a significant deviation from the line at the large ΔT region. This is probably because the nucleation is not the ratedetermining step anymore due to too small nucleation barrier under such a large ΔT. We can discuss the nucleation mechanism of lamellar thickening on the basis of geometrical consideration. As schematically shown in Figure 4, a round double-thickening domain (with a radius of R) grows from a mother lamella (with the thickness of l). The double-thickening could occur with the equal probability on the other side of the mother lamella, but that is a rare event as well in the small simulation system. Usually, double-thickening ends only at one side of the mother lamella. So we discuss only the thickening on one side. The free energy change of such a thickening process is ΔF = ΔFs , l + ΔFs , e + ΔFbulk = 2πRlσl + πR2Δσe + πR2lΔf

(6)

(7)

The first term corresponds to the gain of lateral surface free energy due to the newly created lateral surface of the thickened domain, where σl represents the lateral surface free energy per unit area. The second term corresponds to the fold-end surface

The temperature dependence of the nucleation rates is shown in Figure 3b. One can see that the nucleation rate decreases with the increase of temperature. For a better study of such a nucleation process, the nucleation rates were plotted 167

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Figure 4. Schematic illustration of a doubly thickened domain on the basal plane of a lamellar crystal with thickness l.

When the folded chains are extending in the lamellar crystal, they make stemlike vacancy in the crystal, which can be immediately filled by the surrounding amorphous chains in the melt, raising the crystallinity. Figure 5 demonstrates many blue

Figure 5. Snapshot of the extended-chain lamellar crystal in the melt phase. The yellow chains belong to the originally once-folded lamellar crystal before annealing, while the blue chains represent those amorphous chains before annealing, which are absorbed into the crystalline phases during crystal thickening. Only the crystalline bonds are drawn for clarity.

filled chains in the thickened crystal. A vivid movie can be found in the Supporting Information. Note that the decrease of bulk free energy is much larger than the decrease of fold-end surface free energy upon crystal thickening |ΔFbulk| ≫ |ΔFs , e| Figure 3. (a) Time evolution curve of crystallinity during annealing of once-folded-chain lamellar crystals at T = 4.58 (grown at T = 4.50 with Ep/Ec = 1 and Ef/Ec = 0.02). The onset Monte Carlo cycles of crystal thickening defined the incubation period of nucleation, shown as the crossover of two extrapolation lines. (b) T-dependence of the nucleation rates of lamellar thickening in the melt phase. At each temperature, the reported nucleation rate was averaged over more than 30 times of individual simulations. (c) Logarithmic nucleation rates versus 1/T/ΔT. The linear relationship guided by a straight line demonstrates a secondary-nucleation-type temperature dependence.

Therefore, the discontinuous thickening in the melt phase obeys a bulk-free-energy-driven nucleation-controlled mechanism. Since the initiation of discontinuous thickening follows a nucleation-controlled mechanism, the nucleation rate (I) can be expressed as32 ⎛ ΔF ⎞ I ∝ exp⎜ − c ⎟ ⎝ k bT ⎠

(10)

where ΔFc is the free energy barrier for nucleation due to the free energy penalty of lateral surfaces. By solving the minimum free energy with respect to R in eq 7, such a free energy barrier ΔFc can be found as proportional to ΔT−1, as given by

free energy change, where Δσe represents the change from the fold-end to the extended-end surface free energy per unit area. This term favors crystal thickening. The third term corresponds to the bulk free energy change, where Δf represents the bulk free energy change per unit volume upon crystallization,31 and Δf ∝ −ΔT

(9)

ΔFc =

(8)

πlσl 2 ∝ ΔT −1 −Δf

(11)

Introducing this result into eq 10, we can obtain 168

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1 (12) T ΔT which explains our simulation results for the initiation of crystal thickening in the melt phase (Figure 3c). We also traced the lateral spreading rates of the thickened crystal after nucleation. The linear lateral spreading rate (Ie) is defined as the averaged slope on the time-evolution curves of the thickening front along eight directions (four lattice axes plus four face diagonals) from the center of the thickened domain. The lateral spreading rates increase with ΔT (Figure 6), similar to the conventional secondary-nucleation-controlled ln I ∝ −

Figure 6. T-dependence of the logarithmic lateral spreading rates Ie for the thickened domain on the lamellar crystal of LMW polymers in the melt. The straight line is drawn to guide eyes. Figure 7. Snapshots of the lamellar thickening process of LMW polymers (chain length 16 monomers for initially once folding) in ultrathin films. Blue chains represent amorphous bonds, while yellow ones represent crystalline bonds. (a) The ultrathin film of bulk LMW polymers with a once-folded-chain template (stem length 8 oriented along the vertical direction) and the same height of the thin film in a 64 × 64 × 32 box. (b) Once-folded-chain lamellar crystal grown after 290 000 MCc at T = 4.30 with Ep/Ec = 1 and Ef/Ec = 0, and with B1/Ec = 0.3 and B2/Ec = 0 to show only a repulsive force from the top-surface air. (c) A thickened domain emerging on the top surface of the lamellar crystal after annealing over 600 000 MCc at T = 0.455 with Ep/Ec = 0.1 and Ef/Ec = 0.

lamellar crystal growth. Such a result of spreading rates agree well with Kovacs and co-worker’s observations on the discontinuous thickening of integer-folded PEO crystals.18,19 C. Discontinuous Thickening of Monolayer OnceFolded Short-Chain Crystals on a Solid Substrate. In the third case, the monolayer crystal of LMW polymers is usually prepared in dilute solutions or thin films and be thickened on a solid substrate. Similar like in the previous simulations of thinfilm single crystals,33 we placed 1024 chains, each containing 16 monomers (in order to model LMW polymers), in a 64 × 64 × 8 cubic lattice box with periodic boundary conditions along xaxis and y-axis. The xy plane at z = 0 can be regarded as a solid substrate on which polymers are supported. The film thickness was eight lattice sites comparable to the stem lengths of oncefolded 16-mer chains (eight lattice sites as well). In addition, we set a template formed by one layer of once-folded chains to initiate crystal growth (see Figure 7a). Once-folded-chain lamellar crystals were initiated by the template from the relaxed bulk polymers at T = 4.30, with Ep/Ec = 1 and Ef/Ec = 0 (see Figure 7b). We set B1/Ec = 0.3 to make a repulsive force from the above air with an extended z-axis boundary up to 32, and B2/Ec = 0 at the neutrally repulsive substrate at z = 0. These parameters appeared good enough to stabilize the film for avoiding dewetting. The crystal absorbed all the polymer chains. To accelerate thickening on a solid substrate, we further set Ep/Ec = 0.1 for a more rigid chain. We removed the mobile restriction for those chains on the template and annealed the crystal at an elevated temperature T = 0.455 (corresponding to 4.55 with Ep/Ec = 1). We only observed the initiation of the thickened domain upon annealing (see Figure 7c). The subsequent spreading process of the thickened domains is still too slow to be able to trace within the limited time window of our simulations.

We estimated the incubation period for the initiation of thickened domains. We defined the incubation period ⟨ti⟩ as the required Monte Carlo cycles when the number of thickened crystalline stems exceeds five during isothermal thickening. Here, the lengths of thickened crystalline stems must be larger than 11 bonds. The nucleation rate ⟨I⟩ was represented by the reciprocal of the incubation period. Each result was averaged over 60 individual simulations. We plotted the nucleation rate (I) as a function of temperature in Figure 8a. The nucleation rate increases with temperature. The temperature dependence of the nucleation rates in the ultrathin film appears as opposite to that in the melt phase. For a better study of such a nucleation process, the logarithmic nucleation rates were plotted as a function of T−1, as shown in Figure 8b. A good linear relationship implies that the thickening rates follow an Arrhenius-type temperature dependence. When folded chains are extending on a solid substrate, nearly no amorphous chains are available to fill the stem vacancy due to mass conservation. The vacancy has to be removed to the edges of the crystal or to be accumulated in the holes inside the crystal. The rough edges and holes have been observed in the 169

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conclusion, the discontinuous thickening on a solid substrate exhibits a surface-free-energy-driven nucleation-controlled mechanism.

IV. CONCLUSION Our dynamic Monte Carlo simulations of lattice linear polymers reproduced the controversial experimental observations on the kinetics of isothermal thickening of lamellar crystals, which could be classified into three typical cases. The first case is the basic continuous thickening via sliding diffusion of folded polymer chains in the monolayer lamellar crystal. The thickness exhibits a logarithmic time dependence, which results in a wedge-shaped growth profile at the lateral surface of lamellar crystals. The thickening rate increases with temperature because the thickening process incurs an activation energy barrier for chain-sliding diffusion. The second case is the discontinuous thickening of monolayer crystal of the integer-folded chains in the melt. It exhibits a bulk-free-energy-driven nucleation-controlled mechanism because the amorphous chains can add into lamellar crystals upon thickening to harvest crystallinity. Both the nucleation rates and the lateral spreading rates of the thickened domain incur a temperature-sensitive barrier, resulting in higher rates at lower temperatures. The third case is again the discontinuous thickening of monolayer crystal of integer-folded chains but on a solid substrate. It exhibits a surface-free-energy-driven nucleationcontrolled mechanism because no more amorphous chains can add into lamellar crystals upon thickening. The nucleation rates incur a temperature-insensitive barrier, resulting in higher rates at higher temperatures. Under the facilitation of molecular simulations of lamellar thickening with a proper classification of experimental observations, we have achieved a systematic understanding to various microscopic mechanisms of monolayer lamellar crystal thickening via chain-sliding diffusion. The concerned cases above are mainly eligible to those highly mobile polymers in the crystal, such as PE and PEO. We will pursue our investigations to the lamellar thickening via melting−recrystallization, specifically for those immobile polymers in the crystal, in order to gain further insights into the annealing behaviors of polymer crystals.

Figure 8. (a) Temperature dependences of thickening nucleation rates of LMW polymer lamellae observed in ultrathin films. At each temperature, the nucleation rate is averaged over more than 60 times of individual simulations. (b) Thickening nucleation rate shows an Arrhenius-type temperature dependence. The straight line is drawn to guide eyes.

thickened lamellar crystals by AFM.11−15 Since the volume of the crystalline phase is nearly fixed, and the bulk free energy varies little during crystal thickening on a solid substrate, the decrease of the fold-end surface free energy becomes the dominant factor that drives lamellae thickening, as given by |ΔFbulk| ≪ |ΔFs , e|

(13)

■

In this case, the nucleation barrier becomes πl 2σl 2 ΔFc = −Δσe

ASSOCIATED CONTENT

S Supporting Information *

A movie showing the crystal thickening from the melt induced by the template layer at T = 4.50; each frame was recorded in every 1000 MCc; only those bonds containing more than 15 parallel neighbors are drawn as the crystalline bonds in tiny cylinders; the newly inserted bonds are shown in blue. This material is available free of charge via the Internet at http:// pubs.acs.org.

(14)

We assume that both Δσe and σl are not sensitive to the temperature change,34 and then ΔFc is approximately a constant. Thus, the logarithmic nucleation rate is dominated by the nucleation barrier insensitive to the temperature change, appearing as proportional to the reciprocal of annealing temperature, as given by the Arrhenius-type temperature dependence 1 ln I ∝ − (15) T which explains the simulation results shown in Figure 8b and appears consistent with both Wang’s26 and Chen’s27 experimental observations. The similar results were also confirmed by the time−temperature Ginzburg−Landau equation simulations of such a nucleation process by introducing a potential energy barrier at the fold-end surface upon thickening.27 In

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank Prof. Günter Reiter at Freiburg University and Dr. Yu Ma at Donghua University for helpful discussions. The financial 170

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

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