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Correlating Polymer Crystals via Self-Induced Nucleation Hui Zhang,1 Muhuo Yu,2 Bin Zhang,1 Renate Reiter,1,3 Maximilian Vielhauer,3,4 Rolf Mülhaupt,3,4 Jun Xu,5 and Günter Reiter1,3,* 1
Physikalisches Institut, Albert-Ludwigs-Universität, 79104 Freiburg, Germany State Key Laboratory for Modification of Chemical Fibers and Polymer Materials, College of Materials Science and Engineering, Donghua University, 201620 Shanghai, China 3 Freiburg Materials Research Center (FMF), Albert-Ludwigs-Universität, 79104 Freiburg, Germany 4 Institute for Macromolecular Chemistry, Albert-Ludwigs-Universität, 79104 Freiburg, Germany 5 Institute of Polymer Science and Engineering, Department of Chemical Engineering, School of Materials Science and Technology, Tsinghua University, 100084 Beijing, China (Received 26 March 2014; published 9 June 2014) 2
Crystallizable polymers often form multiple stacks of uniquely oriented lamellae, which have good registry despite being separated by amorphous fold surfaces. These correlations require multiple synchronized, yet unidentified, nucleation events. Here, we demonstrate that in thin films of isotactic polystyrene, the probability of generating correlated lamellae is controlled by the branched morphology of a single primary lamella. The nucleation density ns of secondary lamellae is found to be dependent on the width w of the branches of the primary lamella such that ns ∼ w−2 . This relation is independent of molecular weight, crystallization temperature, and film thickness. We propose a nucleation mechanism based on the insertion of polymers into a branched primary lamellar crystal. DOI: 10.1103/PhysRevLett.112.237801
PACS numbers: 81.10.Aj, 61.41.+e, 68.55.-a
Chemically connecting crystallizable monomers into a chainlike linear polymer does not stop them from crystallizing. However, the connectivity of the units, together with a large number of possible conformations of the polymer chains, significantly affects the kinetics of the crystallization process [1–4]. The resulting metastable crystal is typically composed of crystalline sequences forming a lamella of the order of 10 nm in thickness, sandwiched between two amorphous layers formed by loops, folds, and ends of the very same chains [5–7]. Such amorphous layers represent a barrier for crystal growth in directions normal to the amorphous fold surface. Three-dimensional polymer samples (bulk) are expected to consist of independently nucleated, and thus randomly distributed, quasi-two-dimensional lamellar crystals. However, large stacks of intimately linked lamellae, all parallel to each other, are often observed experimentally for crystallization from supersaturated solutions [7–13] and from undercooled melts [3,7,14–17]. Even in spherulites, parallel lamellae, rather than randomly oriented ones, represent the rule [7]. These findings suggest that a fundamental mechanism exists which allows us to orient many lamellae parallel to each other. Even more surprisingly, plenty of evidence indicates that in such a stack all lamellae are in crystallographic registry [7,9,12,13,17]. Scattering patterns from such three-dimensional crystalline superstructures have signatures of single crystals [3,7,12,13,17]. Taking the separating amorphous interlayers into account, the growth of each lamella has to be initiated individually. Correlated lamellar stacks, therefore, require a correlated nucleation mechanism. In search for such a mechanism, we 0031-9007=14=112(23)=237801(5)
examine the physical parameters allowing us to induce nucleation on the fold surface of a single primary lamella. In particular, we explore relations between the morphology of the primary lamella characterized by the side branch width w and the nucleation density ns of secondary lamellae. Because they grow in thin films, only a few polymers are able to diffuse on the fold surface of a monolamellar crystal [18], making it extremely unlikely to observe any homogeneous nucleation event [19,20]. Nevertheless, as will be shown, the formation of a multitude of stacked lamellae can be clearly observed. Thus, we have to search for a templated nucleation mechanism initiating the correlated formation of many crystalline lamellae on top of amorphous fold surfaces. For various polymers, multiple stacks of correlated lamellae have been studied by various techniques such as transmission electron microscopy, x-ray scattering, and atomic force microscopy (AFM) [1,3,7–17,21]. However, what was primarily obtained were the averaged parameters of the final stacks, which did not provide clear hints to the mechanism leading to their formation. Moreover, it was not possible to identify if all lamellae were nucleated simultaneously or in a temporal sequence of nucleation events. In order to unveil the underlying nucleation mechanism, we have chosen large-scale single crystals in a system where competing homogeneous nucleation was extremely rare, allowing for an unambiguous relation between morphology and polymer orientation. Moreover, crystallization temperatures close to melting points have been chosen in order to avoid perturbations by growth front nucleation [22]. Under such conditions, we obtained dendritic or faceted single crystals rather than spherulitic polycrystallline domains [23].
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Isotactic polystyrene (i-PS), a well-investigated polymer [21,24–33], is characterized in the melt state by a rather high glass transition temperature (T g ∼ 90–100 °C) and low nucleation and crystal growth rates. We used molecular weights (Mw ) of 7100, 13 000, and 48 000 g=mol, corresponding to N ¼ 69, 125, and 462 numbers of monomers, with polydispersities of 1.9, 1.9, and 2.3, respectively. The average maximum length L of the polymers in the fully extended state is L ¼ lu × N ¼ 15, 25, and 102 nm, respectively, with lu ¼ 1=3c ¼ 0.22 nm, where c is the length of the c-axis vector of the unit cell [24]. Films with a thickness ranging from ∼5 to 100 nm were prepared at room temperature by spin coating dilute cyclohexanone solutions onto UV-ozone-cleaned silicon wafers. The solutions were first heated to 150 °C and then filtered. The film thickness was controlled via polymer concentration and was determined by ellipsometry. All i-PS films were first annealed for 3 min on a hot stage purged with nitrogen at temperatures above the respective melting points, followed by a rapid quench to the desired temperature for isothermal crystallization. After crystallization, all samples were quenched to room temperature. The resulting crystal morphologies were investigated by AFM in ambient atmosphere. In the present study, we performed systematic crystallization experiments in ultrathin i-PS films with a focus on crystal morphology in relation to the growth process. AFM topographic images as shown in Fig. 1 clearly demonstrate the hexagonal shape of the crystals, reflecting the symmetry of the crystal unit cell [34]. Previous work [28,31] based on electron diffraction identified the molecular order in such single crystals, with polymer chains being oriented perpendicular to the surface of the substrate. In Fig. 1, we (a)
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want to draw attention to the additional, i.e., “secondary” lamellar crystals formed on top of the amorphous fold surface of the faceted hexagonal lamella at the bottom. Interestingly, during growth, these secondary lamellae could either start immediately at the center [Fig. 1(a)] or, much more frequently, at a later stage off center [Fig. 1(c)] of the bottom crystal. From the discrete and equidistant steps detected in AFM height profiles, we conclude that the orientation of the secondary lamellae was flat-on, i.e., parallel to the underlying lamella. From the fact that all lamellae have a unique orientation of their hexagonal envelopes, we can conclude that all secondary lamellae were in registry with the bottom lamella [3,12,13,17]. This observation suggests that the formation and orientation of all secondary lamellae were directed by the initially formed bottom lamella. Assuming a constant probability of homogenous nucleation, the number of nucleation events is proportional to the volume, i.e., the number of available polymers [35]. Therefore, in a thin film, many fewer nuclei are formed than in a large bulk sample [35]. In addition, the high conformational entropy [4] of polymer chains is partially responsible for the extremely low probability of homogeneous nucleation in polymer systems. In particular, for i-PS, we never observed any nucleation events over distances of many hundred micrometers around a growing crystal. Thus, it is surprising that the nucleation density ns for secondary lamellae on the amorphous fold surface of a lamellar crystal was many orders of magnitude higher than that on the surrounding thin film. In addition, a significant dependence on both crystallization temperature T c and the initial film thickness h (see Fig. 2) was observed. This implies that ns depends on crystal morphology, which is controlled by growth rate GðT c ; hÞ [18,31,36,37]. Thus ns , i.e., the number of secondary lamellae per unit area formed on the fold surface of the underlying primary lamella, was deduced from the
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FIG. 1 (color online). AFM topographic images showing stacks of lamellae of i-PS (Mw ¼ 13 000 g=mol) single crystals. In (a) the stack is initiated at the center and in (c) off center of the underlying crystal. The crystals were grown from a (a) 20 nm thick film at 195 °C and a (c) 16 nm thick film at 190 °C. (b) and (d) show the corresponding three-dimensional height images cut along the black dotted lines indicated in (a) and (c). The sizes of the images are (a) 16 × 16 μm2 and (c) 14 × 14 μm2 .
FIG. 2 (color online). Series of AFM topographic images of i-PS (M w ¼ 13 000 g=mol) single crystals indicating the dependence of the probability of forming secondary lamellae on crystallization temperature T c and film thickness h. Top row: h constant (14 nm) and increasing T c from (a) 150 °C, (b) 160 °C, and (c) 170 °C to (d) 180 °C leads to a decreasing number of secondary lamellae. Bottom row: constant T c of 160 °C and increase in h from (e) 12 nm, (f) 14 nm, and (g) 16 nm to (h) 18 nm leads to an increasing number of secondary lamellae. The scale bar represents 10 μm.
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λ ∼ ðl × d0 Þ1=2 :
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In addition, Langer and Müller-Krumbhaar [41] have proposed that λ is proportional to the radius of curvature ρ at the growth tips of a crystal. According to Langer [39], ρ is also proportional to the critical radius rcr for nucleation. This means that strongly curved growth fronts with ρ < rcr are not stable and will remelt [42]. Only branches with ρ ≥ rcr can grow. Based on a marginal stability hypothesis [41], one obtains ρ2 G ∼ ld0 G ¼ const:;
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where G is the growth rate of the crystal [Fig. 3(a)]. Hence, the parameters rcr ; ρ; λ, and w are all proportional to each other and represent a characteristic length scale of the crystal morphology. ρ can be determined by a parabolic fitting to the curvature of the growth tip (see Fig. S3 in [43]), measured, for example, by AFM (some crystals are shown in Fig. S1 [43]) [39,44,45]. The results of our systematic studies on the dependence of G, ρ, and w on T c , and the products of ρ2 G and w2 G for different values of h and M w , are displayed in Fig. 3. Data for films h > 12 nm are not displayed because it was difficult to determine w. The branches of the dendrites were not distinguishable, as the gaps between side branches were filled. Both ρ and w increased in a similar fashion with crystallization temperature T c [Fig. 3(b)]. Crystals grown in thinner films showed larger values of ρ and w, related to a slower growth rate [23] and an increased diffusion length l [31]. At high T c , larger values of d0 , corresponding to a higher probability of desorption of attached molecules [36], allowed for larger values of ρ and w of the crystals. Consistent
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resulting patterns. For a constant film thickness of h ¼ 14 nm [Figs. 2(a)–2(d)], ns clearly decreased with increasing T c . However, for a constant T c ¼ 160 °C [Figs. 2(e)–(h)], ns increased significantly with increasing h. As h decreased below the thickness LB ðT c ; hÞ of the bottom lamella, ns decreased to approximately zero. In order to establish a relation between ns and the morphology of the primary lamella, it is necessary to identify characteristic morphological parameters. For the limited amount of available polymers in very thin films (h < LB ), the growing primary lamella clearly exhibited a branched morphology [38–41]. The growth process was only nucleated once; i.e., no additional nucleation events occurred, e.g., at the growth front [22]. Thus, all side branches were related to the same nucleus and thus had the unique orientation of the crystal unit cell [3,28,31,34]. Based on the description put forward by Mullins and Sekerka [38–40], the wavelength λ of the branching instability is related to the width w of the resulting side branches. λ characterizes the crystal pattern and is proportional to the geometric mean of the diffusion length l and the capillary length d0 [38–40],
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FIG. 3 (color online). Dependence on crystallization temperature of (a) growth rate G (the velocity at which the main tip of the underlying crystal advanced); (b) tip radius of curvature ρ (solid) and width of the side branches w (open) of the underlying dendritic crystal; (c) the products of w2 G (open) and ρ2 G (solid). All graphs contain data for different molecular weights [7100 g=mol (stars), 13 000 g=mol (diamonds, circles and squares), and 48 000 g=mol (pentagons)] and different film thicknesses [8 nm (stars and diamonds), 9 nm (pentagons), 10 nm (circles), and 12 nm (squares)].
with theoretical predictions [41] and previous experiments [23], for the temperature range above the maximum crystal growth rate, the products of w2 G and ρ2 G were found to be approximately constant [Fig. 3(c)]. At lower temperatures, on approaching the glass transition, the parameters ρ, w, and G, and thus also ρ2 G and w2 G, decreased [1,7]. In Fig. 4(a), we present the dependence of ns on T c , for different values of h and Mw . For all samples, ns decreased with T c and increased with h. For a given T c , ns increased with Mw . Thus, opposite to the variation of w and ρ with T c , the value of ns decreased with increasing T c . Figure 4(b) clearly shows that ns is obviously affected by morphological parameters of the underlying primary lamellar crystal, suggesting the following relation: ns ∼ w−2 :
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This relation is independent of h, T c , and M w . Thus, the probability for nucleating secondary lamellae is obviously correlated with the morphology of the primary lamella, characterized by w. The exponent of −2 in Eq. (3) implies that the nucleation density is constant if normalized by an area proportional to w2 (see Fig. S5 [43]). In other words, the smaller the area of a branch, i.e., the higher the branching density, the higher the number of secondary lamellae. We identified positions in between the branches as sites for nucleation on the amorphous fold surface. In particular along the narrow junction lines where branches merge, only parts of polymer chains can be inserted in the crystal [46]. The remainders of the chains are left outside, generating
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FIG. 4 (color online). Density ns of secondary lamellae as a function of (a) crystallization temperature T c and (b) side branch width w, shown for samples of different molecular weights [7100 g=mol (stars), 13 000 g=mol (diamonds, circles, squares), and 48 000 g=mol (pentagons)] and film thicknesses [8 nm (stars and diamonds), 9 nm (pentagons), 10 nm (circles), and 12 nm (squares)]. Graph (a) indicates that ns decreased rapidly with increasing crystallization temperature for all samples. While the slope of this decay is similar for all samples, the curves are shifted to higher temperatures for higher Mw. In graph (b), all curves approximately superimpose. The solid line is a linear fit of all secondary lamellar densities ns as a function of side branch width w, yielding a slope of −2 with an error bar of 0.1.
locally a high density of chain segments. These segments are anchored in the gap and can form a bridge between lamellae [12], as visualized at a molecular level by AFM [47]. The chains anchored in the primary lamella may act as an attractor for other chains diffusing on the fold surface, potentially enhancing the local area density of chains and thus the nucleation probability [48]. Assuming a coupling between the crystalline stems inserted in the narrow gap and the amorphous polymer segments outside the gap, we may anticipate a mechanism of ordering and orientation propagating from the crystalline to the amorphous chain segments. Such a propagation may lead to the nucleation of correlated secondary lamellae [12,13,17]. The two lamellae necessarily share at least a few polymer chains in common, which have the potential to have identically oriented crystalline chain segments and which thus may cause these lamellae to be in registry [12,13,17]. This concept of insertion-induced nucleation is supported by microscopy observations indicating that secondary lamellae formed almost exclusively along junction lines (see Fig. 5 and Fig. S4 [43]). Thus, ns is expected to be proportional to the number density of junction lines, which is determined by w. For a two-dimensional dendritic crystal with regular side branches, the number density of the junction lines is proportional to 1=w2 and, thus, ns ∼ w−2 . From the rather small values of ns shown in Fig. 4, we may deduce that successful nucleation did not occur at each junction. For example, for h ¼ 12 nm, we can conclude that secondary lamellae only formed at about 3% of junctions characterized by an area w2 . Moreover, in films thinner than the crystalline lamella (h < LB ), ns was even smaller because only a small fraction of the available polymers succeeded in diffusing onto the fold surface. Growth of new lamellae on fold surfaces requires the
FIG. 5 (color online). AFM height image of an i-PS (M w ¼ 13 kg=mol) single crystal (a) shows the start of secondary lamellae formation at a junction of the branches of the underlying dendrite. The crystal was grown from an 8-nm thick film at 150 °C. The corresponding three-dimensional height image of the area marked by a red rectangle in image (a) indicates that the secondary lamella grew at a junction of the underlying primary lamella (b). The sizes of the images are 32 × 32 μm2 and 2 × 1.5 μm2 for (a) and (b), respectively.
transport of polymer chains to nucleation sites. For thin films with h < LB , transport occurs first essentially within the quasi-two-dimensional film or on the fold surface of the growing lamella [18,31,37]. Polymer molecules from the surrounding reservoir diffuse across a depletion zone towards the crystal, where they attempt to attach. For all film thicknesses, including h < LB , polymer chains that attached to the growth front have a nonzero detachment probability, which increases with T c [36]. These detached polymer chains can diffuse backward into the supplying reservoir. Alternatively, these polymer chains can also diffuse upward onto the fold surface of the growing lamella, forming a reservoir of free molecules for crystal growth on the amorphous fold surface. For h > LB, a larger number of polymers diffused on the fold surface and led to a higher nucleation density via filling the gaps between side branches, e.g., a process similar to the one proposed in [49] for crystals of small organic molecules. For much thicker films (i.e., much higher supply of polymers) or at lower T c (higher nucleation density of secondary lamellae), coalescence between the growing secondary lamellae, and thus the formation of a continuous lamellar layer on top of the bottom one, occurred [50]. Interestingly, even for much thicker films, perfect registry between all lamellae, represented by a unique orientation of all hexagonal envelopes, was observed. Such stacking of uniquely oriented lamellae opens up the possibility for crystallizing polymers in a single-crystal-like fashion in the direction normal to the fold surface, i.e., across amorphous interphases [33]. In summary, a clear correlation exists between the morphology of the underlying primary lamellar crystal and the probability of nucleating lamellar stacks. Based on a morphology-induced nucleation mechanism, order can propagate across amorphous interlayers. By stacking polymer lamellar crystals in registry, three-dimensional polymer single crystals are possible, although they contain amorphous interlayers.
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The authors are grateful to Professor Gert Strobl, Professor Murugappan Muthukumar, Professor Wenbing Hu, Mrs. Barbara Heck, Dr. Florian Spieckermann, and Dr. Shu Zhu for their helpful discussions. We also would like to acknowledge financial support from the Sino-German Center for Research Promotion and the German Science Foundation. M. Y. would like to acknowledge the National Program of China on Key Basic Research Project No. 2011CB606100. H. Z. and B. Z. wish to thank the China Scholarship Council (CSC), which financially supported their Ph.D. study at the University of Freiburg.
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[email protected]‑freiburg.de [1] G. Strobl, The Physics of Polymers: Concepts for Understanding Their Structures and Behavior (Springer, Berlin, Heidelberg, 2007). [2] G. Reiter and G. Strobl, Progress in Understanding of Polymer Crystallization (Springer, Berlin, Heidelberg, 2007). [3] G. Reiter, Chem. Soc. Rev. 43, 2055 (2014). [4] P. Welch and M. Muthukumar, Phys. Rev. Lett. 87, 218302 (2001). [5] A. Keller and S. Z. Cheng, Polymer 39, 4461 (1998). [6] J. P. K. Doye and D. Frenkel, Phys. Rev. Lett. 81, 2160 (1998). [7] B. Wunderlich, Macromolecular Physics, Volume 1: Crystal Structure, Morphology, Defects (Academic Press, Mishawaka, 1973). [8] D. Reneker and P. Geil, J. Appl. Phys. 31, 1916 (1960). [9] P. H. Geil, Polymer Single Crystals (Interscience, New York, 1964), p. 214. [10] H. D. Keith, R. G. Vadimsky, and F. J. Padden, J. Polym. Sci., Part A-2 8, 1687 (1970). [11] H. Keith and F. Padden, J. Polym. Sci., Part B: Polym. Phys. 25, 2371 (1987). [12] H. Keith and W. Chen, Polymer 43, 6263 (2002). [13] M. T. Casas, J. Puiggalí, J.-M. Raquez, P. Dubois, M. E. Córdova, and A. J. Müller, Polymer 52, 5166 (2011). [14] A. Toda and A. Keller, Colloid Polym. Sci. 271, 328 (1993). [15] A. Toda, M. Okamura, M. Hikosaka, and Y. Nakagawa, Polymer 46, 8708 (2005). [16] M. Franke and N. Rehse, Macromolecules 41, 163 (2008). [17] S. Hong, W. J. MacKnight, T. P. Russell, and S. P. Gido, Macromolecules 34, 2398 (2001). [18] G. Reiter and J.-U. Sommer, J. Chem. Phys. 112, 4376 (2000). [19] M. V. Massa and K. Dalnoki-Veress, Phys. Rev. Lett. 92, 255509 (2004). [20] E. Woo, J. Huh, Y. G. Jeong, and K. Shin, Phys. Rev. Lett. 98, 136103 (2007). [21] G. Strobl, Prog. Polym. Sci. 31, 398 (2006). [22] L. Gránásy, T. Pusztai, T. Börzsönyi, J. A. Warren, and J. F. Douglas, Nat. Mater. 3, 645 (2004). [23] N. Grozev, I. Botiz, and G. Reiter, Eur. Phys. J. E 27, 63 (2008). [24] G. Natta, P. Corradini, and I. Bassi, Nuovo Cimento 15, 68 (1960). [25] J. Boon, G. Challa, and D. W. Van Krevelen, J. Polym. Sci., Part A-2 6, 1791 (1968).
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[26] T. Suzuki and A. J. Kovacs, Polym. J. (Tokyo, Jpn.) 1, 82 (1970). [27] Y. K. Godovsky and G. Slonimsky, J. Polym. Sci., Polym. Phys. Ed. 12, 1053 (1974). [28] K. Izumi, P. Gan, A. Toda, H. Miyaji, and Y. Miyamoto, Jpn. J. Appl. Phys. 31, L626 (1992). [29] S. Sawamura, H. Miyaji, K. Izumi, S. J. Sutton, and Y. Miyamoto, J. Phys. Soc. Jpn. 67, 3338 (1998). [30] Y. Miyamoto, Y. Tanzawa, H. Miyaji, and H. Kiho, Polymer 33, 2496 (1992). [31] K. Taguchi, H. Miyaji, K. Izumi, A. Hoshino, Y. Miyamoto, and R. Kokawa, Polymer 42, 7443 (2001). [32] K. Taguchi, H. Miyaji, K. Izumi, A. Hoshino, Y. Miyamoto, and R. Kokawa, J. Macromol. Sci., Part B: Phys. 41, 1033 (2002). [33] K. Izumi, P. Gan, M. Hashimoto, A. Toda, H. Miyaji, Y. Miyamoto, and N. Yoshitsugu, Crystal Growth of Polymers in Thin Films, Advances in the Understanding of Crystal Growth Mechanisms, edited by T. Nishinaga, K. Nishioka, J. Harada, A. Sasaki, and H. Takei (Elsevier, Amsterdam, 1997), p. 337. [34] J. Xu, Y. Ma, W. Hu, M. Rehahn, and G. Reiter, Nat. Mater. 8, 348 (2009). [35] M. V. Massa, J. L. Carvalho, and K. Dalnoki-Veress, Phys. Rev. Lett. 97, 247802 (2006). [36] G. Reiter and J.-U. Sommer, Phys. Rev. Lett. 80, 3771 (1998). [37] W. Bi, J. S. Teguh, and E. K. L. Yeow, Phys. Rev. Lett. 102, 048302 (2009). [38] W. W. Mullins and R. Sekerka, J. Appl. Phys. 35, 444 (1964). [39] J. Langer, Rev. Mod. Phys. 52, 1 (1980). [40] R. F. Sekerka, J. Cryst. Growth 264, 530 (2004). [41] J. Langer and J. Müller-Krumbhaar, J. Cryst. Growth 42, 11 (1977). [42] R. F. Sekerka, J. Cryst. Growth 128, 1 (1993). [43] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.237801 for the morphologies of primary dendritic crystal of various i-PS samples differing in molecular weights, the procedure for determining the side branch width and the radius of curvature of the tip of primary dendritic crystals, the nucleation sites of secondary lamellae on the fold surface of the underlying primary lamella and some common features of secondary lamellae. [44] G. P. Ivantsov, Dokl. Akad. Nauk SSSR 58, 567 (1947) [Sov. Phys. Dokl. 58, 567 (1947)]; D. Temkin, Acta Mater. 53, 2733 (2005). [45] J.-M. Debierre, A. Karma, F. Celestini, and R. Guérin, Phys. Rev. E 68, 041604 (2003). [46] P.-G. de Gennes, J. Chem. Phys. 55, 572 (1971). [47] N. Mullin and J. K. Hobbs, Phys. Rev. Lett. 107, 197801 (2011). [48] J. L. Carvalho and K. Dalnoki-Veress, Phys. Rev. Lett. 105, 237801 (2010). [49] M. Voigt, S. Dorsfeld, A. Volz, and M. Sokolowski, Phys. Rev. Lett. 91, 026103 (2003). [50] A. M. Streets and S. R. Quake, Phys. Rev. Lett. 104, 178102 (2010).
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Correlating Polymer Crystals via Self-Induced Nucleation Hui Zhang,1 Muhuo Yu,2 Bin Zhang,1 Renate Reiter,1,3 Maximilian Vielhauer,3,4 Rolf Mülhaupt,3,4 Jun Xu,5 and Günter Reiter1,3,* 1
Physikalisches Institut, Albert-Ludwigs-Universität, 79104 Freiburg, Germany State Key Laboratory for Modification of Chemical Fibers and Polymer Materials, College of Materials Science and Engineering, Donghua University, 201620 Shanghai, China 3 Freiburg Materials Research Center (FMF), Albert-Ludwigs-Universität, 79104 Freiburg, Germany 4 Institute for Macromolecular Chemistry, Albert-Ludwigs-Universität, 79104 Freiburg, Germany 5 Institute of Polymer Science and Engineering, Department of Chemical Engineering, School of Materials Science and Technology, Tsinghua University, 100084 Beijing, China (Received 26 March 2014; published 9 June 2014) 2
Crystallizable polymers often form multiple stacks of uniquely oriented lamellae, which have good registry despite being separated by amorphous fold surfaces. These correlations require multiple synchronized, yet unidentified, nucleation events. Here, we demonstrate that in thin films of isotactic polystyrene, the probability of generating correlated lamellae is controlled by the branched morphology of a single primary lamella. The nucleation density ns of secondary lamellae is found to be dependent on the width w of the branches of the primary lamella such that ns ∼ w−2 . This relation is independent of molecular weight, crystallization temperature, and film thickness. We propose a nucleation mechanism based on the insertion of polymers into a branched primary lamellar crystal. DOI: 10.1103/PhysRevLett.112.237801
PACS numbers: 81.10.Aj, 61.41.+e, 68.55.-a
Chemically connecting crystallizable monomers into a chainlike linear polymer does not stop them from crystallizing. However, the connectivity of the units, together with a large number of possible conformations of the polymer chains, significantly affects the kinetics of the crystallization process [1–4]. The resulting metastable crystal is typically composed of crystalline sequences forming a lamella of the order of 10 nm in thickness, sandwiched between two amorphous layers formed by loops, folds, and ends of the very same chains [5–7]. Such amorphous layers represent a barrier for crystal growth in directions normal to the amorphous fold surface. Three-dimensional polymer samples (bulk) are expected to consist of independently nucleated, and thus randomly distributed, quasi-two-dimensional lamellar crystals. However, large stacks of intimately linked lamellae, all parallel to each other, are often observed experimentally for crystallization from supersaturated solutions [7–13] and from undercooled melts [3,7,14–17]. Even in spherulites, parallel lamellae, rather than randomly oriented ones, represent the rule [7]. These findings suggest that a fundamental mechanism exists which allows us to orient many lamellae parallel to each other. Even more surprisingly, plenty of evidence indicates that in such a stack all lamellae are in crystallographic registry [7,9,12,13,17]. Scattering patterns from such three-dimensional crystalline superstructures have signatures of single crystals [3,7,12,13,17]. Taking the separating amorphous interlayers into account, the growth of each lamella has to be initiated individually. Correlated lamellar stacks, therefore, require a correlated nucleation mechanism. In search for such a mechanism, we 0031-9007=14=112(23)=237801(5)
examine the physical parameters allowing us to induce nucleation on the fold surface of a single primary lamella. In particular, we explore relations between the morphology of the primary lamella characterized by the side branch width w and the nucleation density ns of secondary lamellae. Because they grow in thin films, only a few polymers are able to diffuse on the fold surface of a monolamellar crystal [18], making it extremely unlikely to observe any homogeneous nucleation event [19,20]. Nevertheless, as will be shown, the formation of a multitude of stacked lamellae can be clearly observed. Thus, we have to search for a templated nucleation mechanism initiating the correlated formation of many crystalline lamellae on top of amorphous fold surfaces. For various polymers, multiple stacks of correlated lamellae have been studied by various techniques such as transmission electron microscopy, x-ray scattering, and atomic force microscopy (AFM) [1,3,7–17,21]. However, what was primarily obtained were the averaged parameters of the final stacks, which did not provide clear hints to the mechanism leading to their formation. Moreover, it was not possible to identify if all lamellae were nucleated simultaneously or in a temporal sequence of nucleation events. In order to unveil the underlying nucleation mechanism, we have chosen large-scale single crystals in a system where competing homogeneous nucleation was extremely rare, allowing for an unambiguous relation between morphology and polymer orientation. Moreover, crystallization temperatures close to melting points have been chosen in order to avoid perturbations by growth front nucleation [22]. Under such conditions, we obtained dendritic or faceted single crystals rather than spherulitic polycrystallline domains [23].
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Isotactic polystyrene (i-PS), a well-investigated polymer [21,24–33], is characterized in the melt state by a rather high glass transition temperature (T g ∼ 90–100 °C) and low nucleation and crystal growth rates. We used molecular weights (Mw ) of 7100, 13 000, and 48 000 g=mol, corresponding to N ¼ 69, 125, and 462 numbers of monomers, with polydispersities of 1.9, 1.9, and 2.3, respectively. The average maximum length L of the polymers in the fully extended state is L ¼ lu × N ¼ 15, 25, and 102 nm, respectively, with lu ¼ 1=3c ¼ 0.22 nm, where c is the length of the c-axis vector of the unit cell [24]. Films with a thickness ranging from ∼5 to 100 nm were prepared at room temperature by spin coating dilute cyclohexanone solutions onto UV-ozone-cleaned silicon wafers. The solutions were first heated to 150 °C and then filtered. The film thickness was controlled via polymer concentration and was determined by ellipsometry. All i-PS films were first annealed for 3 min on a hot stage purged with nitrogen at temperatures above the respective melting points, followed by a rapid quench to the desired temperature for isothermal crystallization. After crystallization, all samples were quenched to room temperature. The resulting crystal morphologies were investigated by AFM in ambient atmosphere. In the present study, we performed systematic crystallization experiments in ultrathin i-PS films with a focus on crystal morphology in relation to the growth process. AFM topographic images as shown in Fig. 1 clearly demonstrate the hexagonal shape of the crystals, reflecting the symmetry of the crystal unit cell [34]. Previous work [28,31] based on electron diffraction identified the molecular order in such single crystals, with polymer chains being oriented perpendicular to the surface of the substrate. In Fig. 1, we (a)
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want to draw attention to the additional, i.e., “secondary” lamellar crystals formed on top of the amorphous fold surface of the faceted hexagonal lamella at the bottom. Interestingly, during growth, these secondary lamellae could either start immediately at the center [Fig. 1(a)] or, much more frequently, at a later stage off center [Fig. 1(c)] of the bottom crystal. From the discrete and equidistant steps detected in AFM height profiles, we conclude that the orientation of the secondary lamellae was flat-on, i.e., parallel to the underlying lamella. From the fact that all lamellae have a unique orientation of their hexagonal envelopes, we can conclude that all secondary lamellae were in registry with the bottom lamella [3,12,13,17]. This observation suggests that the formation and orientation of all secondary lamellae were directed by the initially formed bottom lamella. Assuming a constant probability of homogenous nucleation, the number of nucleation events is proportional to the volume, i.e., the number of available polymers [35]. Therefore, in a thin film, many fewer nuclei are formed than in a large bulk sample [35]. In addition, the high conformational entropy [4] of polymer chains is partially responsible for the extremely low probability of homogeneous nucleation in polymer systems. In particular, for i-PS, we never observed any nucleation events over distances of many hundred micrometers around a growing crystal. Thus, it is surprising that the nucleation density ns for secondary lamellae on the amorphous fold surface of a lamellar crystal was many orders of magnitude higher than that on the surrounding thin film. In addition, a significant dependence on both crystallization temperature T c and the initial film thickness h (see Fig. 2) was observed. This implies that ns depends on crystal morphology, which is controlled by growth rate GðT c ; hÞ [18,31,36,37]. Thus ns , i.e., the number of secondary lamellae per unit area formed on the fold surface of the underlying primary lamella, was deduced from the
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FIG. 1 (color online). AFM topographic images showing stacks of lamellae of i-PS (Mw ¼ 13 000 g=mol) single crystals. In (a) the stack is initiated at the center and in (c) off center of the underlying crystal. The crystals were grown from a (a) 20 nm thick film at 195 °C and a (c) 16 nm thick film at 190 °C. (b) and (d) show the corresponding three-dimensional height images cut along the black dotted lines indicated in (a) and (c). The sizes of the images are (a) 16 × 16 μm2 and (c) 14 × 14 μm2 .
FIG. 2 (color online). Series of AFM topographic images of i-PS (M w ¼ 13 000 g=mol) single crystals indicating the dependence of the probability of forming secondary lamellae on crystallization temperature T c and film thickness h. Top row: h constant (14 nm) and increasing T c from (a) 150 °C, (b) 160 °C, and (c) 170 °C to (d) 180 °C leads to a decreasing number of secondary lamellae. Bottom row: constant T c of 160 °C and increase in h from (e) 12 nm, (f) 14 nm, and (g) 16 nm to (h) 18 nm leads to an increasing number of secondary lamellae. The scale bar represents 10 μm.
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λ ∼ ðl × d0 Þ1=2 :
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In addition, Langer and Müller-Krumbhaar [41] have proposed that λ is proportional to the radius of curvature ρ at the growth tips of a crystal. According to Langer [39], ρ is also proportional to the critical radius rcr for nucleation. This means that strongly curved growth fronts with ρ < rcr are not stable and will remelt [42]. Only branches with ρ ≥ rcr can grow. Based on a marginal stability hypothesis [41], one obtains ρ2 G ∼ ld0 G ¼ const:;
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where G is the growth rate of the crystal [Fig. 3(a)]. Hence, the parameters rcr ; ρ; λ, and w are all proportional to each other and represent a characteristic length scale of the crystal morphology. ρ can be determined by a parabolic fitting to the curvature of the growth tip (see Fig. S3 in [43]), measured, for example, by AFM (some crystals are shown in Fig. S1 [43]) [39,44,45]. The results of our systematic studies on the dependence of G, ρ, and w on T c , and the products of ρ2 G and w2 G for different values of h and M w , are displayed in Fig. 3. Data for films h > 12 nm are not displayed because it was difficult to determine w. The branches of the dendrites were not distinguishable, as the gaps between side branches were filled. Both ρ and w increased in a similar fashion with crystallization temperature T c [Fig. 3(b)]. Crystals grown in thinner films showed larger values of ρ and w, related to a slower growth rate [23] and an increased diffusion length l [31]. At high T c , larger values of d0 , corresponding to a higher probability of desorption of attached molecules [36], allowed for larger values of ρ and w of the crystals. Consistent
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resulting patterns. For a constant film thickness of h ¼ 14 nm [Figs. 2(a)–2(d)], ns clearly decreased with increasing T c . However, for a constant T c ¼ 160 °C [Figs. 2(e)–(h)], ns increased significantly with increasing h. As h decreased below the thickness LB ðT c ; hÞ of the bottom lamella, ns decreased to approximately zero. In order to establish a relation between ns and the morphology of the primary lamella, it is necessary to identify characteristic morphological parameters. For the limited amount of available polymers in very thin films (h < LB ), the growing primary lamella clearly exhibited a branched morphology [38–41]. The growth process was only nucleated once; i.e., no additional nucleation events occurred, e.g., at the growth front [22]. Thus, all side branches were related to the same nucleus and thus had the unique orientation of the crystal unit cell [3,28,31,34]. Based on the description put forward by Mullins and Sekerka [38–40], the wavelength λ of the branching instability is related to the width w of the resulting side branches. λ characterizes the crystal pattern and is proportional to the geometric mean of the diffusion length l and the capillary length d0 [38–40],
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FIG. 3 (color online). Dependence on crystallization temperature of (a) growth rate G (the velocity at which the main tip of the underlying crystal advanced); (b) tip radius of curvature ρ (solid) and width of the side branches w (open) of the underlying dendritic crystal; (c) the products of w2 G (open) and ρ2 G (solid). All graphs contain data for different molecular weights [7100 g=mol (stars), 13 000 g=mol (diamonds, circles and squares), and 48 000 g=mol (pentagons)] and different film thicknesses [8 nm (stars and diamonds), 9 nm (pentagons), 10 nm (circles), and 12 nm (squares)].
with theoretical predictions [41] and previous experiments [23], for the temperature range above the maximum crystal growth rate, the products of w2 G and ρ2 G were found to be approximately constant [Fig. 3(c)]. At lower temperatures, on approaching the glass transition, the parameters ρ, w, and G, and thus also ρ2 G and w2 G, decreased [1,7]. In Fig. 4(a), we present the dependence of ns on T c , for different values of h and Mw . For all samples, ns decreased with T c and increased with h. For a given T c , ns increased with Mw . Thus, opposite to the variation of w and ρ with T c , the value of ns decreased with increasing T c . Figure 4(b) clearly shows that ns is obviously affected by morphological parameters of the underlying primary lamellar crystal, suggesting the following relation: ns ∼ w−2 :
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This relation is independent of h, T c , and M w . Thus, the probability for nucleating secondary lamellae is obviously correlated with the morphology of the primary lamella, characterized by w. The exponent of −2 in Eq. (3) implies that the nucleation density is constant if normalized by an area proportional to w2 (see Fig. S5 [43]). In other words, the smaller the area of a branch, i.e., the higher the branching density, the higher the number of secondary lamellae. We identified positions in between the branches as sites for nucleation on the amorphous fold surface. In particular along the narrow junction lines where branches merge, only parts of polymer chains can be inserted in the crystal [46]. The remainders of the chains are left outside, generating
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FIG. 4 (color online). Density ns of secondary lamellae as a function of (a) crystallization temperature T c and (b) side branch width w, shown for samples of different molecular weights [7100 g=mol (stars), 13 000 g=mol (diamonds, circles, squares), and 48 000 g=mol (pentagons)] and film thicknesses [8 nm (stars and diamonds), 9 nm (pentagons), 10 nm (circles), and 12 nm (squares)]. Graph (a) indicates that ns decreased rapidly with increasing crystallization temperature for all samples. While the slope of this decay is similar for all samples, the curves are shifted to higher temperatures for higher Mw. In graph (b), all curves approximately superimpose. The solid line is a linear fit of all secondary lamellar densities ns as a function of side branch width w, yielding a slope of −2 with an error bar of 0.1.
locally a high density of chain segments. These segments are anchored in the gap and can form a bridge between lamellae [12], as visualized at a molecular level by AFM [47]. The chains anchored in the primary lamella may act as an attractor for other chains diffusing on the fold surface, potentially enhancing the local area density of chains and thus the nucleation probability [48]. Assuming a coupling between the crystalline stems inserted in the narrow gap and the amorphous polymer segments outside the gap, we may anticipate a mechanism of ordering and orientation propagating from the crystalline to the amorphous chain segments. Such a propagation may lead to the nucleation of correlated secondary lamellae [12,13,17]. The two lamellae necessarily share at least a few polymer chains in common, which have the potential to have identically oriented crystalline chain segments and which thus may cause these lamellae to be in registry [12,13,17]. This concept of insertion-induced nucleation is supported by microscopy observations indicating that secondary lamellae formed almost exclusively along junction lines (see Fig. 5 and Fig. S4 [43]). Thus, ns is expected to be proportional to the number density of junction lines, which is determined by w. For a two-dimensional dendritic crystal with regular side branches, the number density of the junction lines is proportional to 1=w2 and, thus, ns ∼ w−2 . From the rather small values of ns shown in Fig. 4, we may deduce that successful nucleation did not occur at each junction. For example, for h ¼ 12 nm, we can conclude that secondary lamellae only formed at about 3% of junctions characterized by an area w2 . Moreover, in films thinner than the crystalline lamella (h < LB ), ns was even smaller because only a small fraction of the available polymers succeeded in diffusing onto the fold surface. Growth of new lamellae on fold surfaces requires the
FIG. 5 (color online). AFM height image of an i-PS (M w ¼ 13 kg=mol) single crystal (a) shows the start of secondary lamellae formation at a junction of the branches of the underlying dendrite. The crystal was grown from an 8-nm thick film at 150 °C. The corresponding three-dimensional height image of the area marked by a red rectangle in image (a) indicates that the secondary lamella grew at a junction of the underlying primary lamella (b). The sizes of the images are 32 × 32 μm2 and 2 × 1.5 μm2 for (a) and (b), respectively.
transport of polymer chains to nucleation sites. For thin films with h < LB , transport occurs first essentially within the quasi-two-dimensional film or on the fold surface of the growing lamella [18,31,37]. Polymer molecules from the surrounding reservoir diffuse across a depletion zone towards the crystal, where they attempt to attach. For all film thicknesses, including h < LB , polymer chains that attached to the growth front have a nonzero detachment probability, which increases with T c [36]. These detached polymer chains can diffuse backward into the supplying reservoir. Alternatively, these polymer chains can also diffuse upward onto the fold surface of the growing lamella, forming a reservoir of free molecules for crystal growth on the amorphous fold surface. For h > LB, a larger number of polymers diffused on the fold surface and led to a higher nucleation density via filling the gaps between side branches, e.g., a process similar to the one proposed in [49] for crystals of small organic molecules. For much thicker films (i.e., much higher supply of polymers) or at lower T c (higher nucleation density of secondary lamellae), coalescence between the growing secondary lamellae, and thus the formation of a continuous lamellar layer on top of the bottom one, occurred [50]. Interestingly, even for much thicker films, perfect registry between all lamellae, represented by a unique orientation of all hexagonal envelopes, was observed. Such stacking of uniquely oriented lamellae opens up the possibility for crystallizing polymers in a single-crystal-like fashion in the direction normal to the fold surface, i.e., across amorphous interphases [33]. In summary, a clear correlation exists between the morphology of the underlying primary lamellar crystal and the probability of nucleating lamellar stacks. Based on a morphology-induced nucleation mechanism, order can propagate across amorphous interlayers. By stacking polymer lamellar crystals in registry, three-dimensional polymer single crystals are possible, although they contain amorphous interlayers.
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The authors are grateful to Professor Gert Strobl, Professor Murugappan Muthukumar, Professor Wenbing Hu, Mrs. Barbara Heck, Dr. Florian Spieckermann, and Dr. Shu Zhu for their helpful discussions. We also would like to acknowledge financial support from the Sino-German Center for Research Promotion and the German Science Foundation. M. Y. would like to acknowledge the National Program of China on Key Basic Research Project No. 2011CB606100. H. Z. and B. Z. wish to thank the China Scholarship Council (CSC), which financially supported their Ph.D. study at the University of Freiburg.
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