Structure Evolution in Directional Crystallization of Polymers under

Article pubs.acs.org/Macromolecules

Structure Evolution in Directional Crystallization of Polymers under Temperature Gradient Akihiko Toda,* Ken Taguchi, and Hiroshi Kajioka Graduate School of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan ABSTRACT: In order to clarify the formation mechanism of polymer spherulites, we have experimentally examined the effect of externally applied gradient field of temperature on the structural evolution of polymer crystallization, especially on the characteristic length scales of the inner structures of nonbanded crystallization of isotactic poly(butene-1) and of banded crystallization of poly(vinylidene fluoride) and polyethylene. The inner structures were strongly influenced by the temperature gradient and suggested the important role of the gradient field of temperature, i.e., of chemical potential in general, at the growth front in the bulk melt. The results are in accordance with our proposal and experimental confirmation on the formation of polymer spherulites based on the instability-driven branching promoted by a self-induced gradient field in the bulk melt.

1. INTRODUCTION Formation of polycrystalline aggregates of polymer crystals, socalled spherulites, requires a space-filling branching with noncrystallographic reorientation of lamellar crystals. In our prior experimental studies, a classical modeling1,2 based on the instability-driven branching has been experimentally examined for the banded spherulites of polyethylene (PE)3,4 and poly(vinylidene fluoride) (PVDF)5 and the nonbanded spherulites of isotactic poly(butene-1) (iPB-1)6,7 and isotactic polystyrene (iPS).8 In the original modeling of Keith and Padden,1 the instability has been assumed to be caused by a self-induced gradient field of impurities, which are excluded from crystal growth front and form the gradient field with the highest concentration at the growth front. This gradient evolves the gradient in the driving force of crystallization with its minimum at the growth front and hence promotes the fingering instability of the growth front for branching. In our prior works, in addition to the possible effects of impurities, we noted the effect of a mechanical gradient field with negative pressure at the growth front, which is inevitably required for the steadystate growth of crystals with melt flow to compensate for the crystal−melt density gap.9 Though the direct experimental confirmation of the gradient field has been unsuccessful for many decades,1,2,10−14 the prediction of the instability-driven branching with the stabilizing effect of surface tension, γ, has been successfully confirmed for the above-mentioned polymers in our recent works.3−8 The critical lamellar width, λ, for branching is determined by the gradient of driving force at the growth front, a, and represented as

λ = 2π(vs γ /a)1/2 λ ∝ (γD/V )1/2 1/2

λ ∝ (γ /ηV )

where vs represents the specific volume of a segment in the crystal. In eqs 2 and 3, the effects of compositional and mechanical gradients are influenced by the crystal growth rate, V, the diffusion coefficient, D, and the viscosity, η. We have experimentally confirmed the temperature dependence of λ following the above prediction for PE,3,4 PVDF,5 iPB-1,6,7 and iPS.8 We have further tried to differentiate those effects of the compositional and mechanical gradients by means of the molecular weight dependences of those coefficients, D and η, for PE4 and iPB-17 and of the decoupling of the related processes, i.e., translational and rotational diffusions of molecules, near the glass transition for iPS.8 The obtained results were in accordance with the mechanical gradient, not with the compositional gradient. Those results were of the supporting evidence for the instability caused by a self-induced gradient field. In the present paper, we take an approach to this issue from different angles with the application of an external field to see the morphological change in the inner structure of polymer spherulites. As the external field, we can think of temperature, flow, and electric and magnetic fields. In the present approach, we apply temperature gradient for the steady-state crystallization with moving interface at a constant speed. Crystallization under temperature gradient has been often utilized for the purpose of directional solidification with oriented crystallites realized by moving of the sample along the temperature gradient at a constant speed.15−26 Under temperature gradient, the solidification spontaneously adjusts the position of the crystal−melt interface to choose the temperature at which crystal grows at the preset speed of moving. In other words, the gradient suppresses the fluctuations of the position of growth front.

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compositional

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© 2011 American Chemical Society

Received: October 2, 2011 Revised: November 30, 2011 Published: December 23, 2011 852

dx.doi.org/10.1021/ma2022088 | Macromolecules 2012, 45, 852−861

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In the following, we introduce the equipment for the directional solidification under temperature gradient, which is specialized for the moving at relatively slow speed in accordance with slow polymer crystallization. Then, the experimental results of the observations are discussed for the nonbanded crystallization of iPB-1 and the banded crystallization of PVDF and PE.

Under temperature gradient, it is well-known that the growth of dendrites of organic and metallic alloys becomes directional and cellular with the formation of a periodic array of dendrites.27−32 Dendrites of alloys are generally formed by a fingering instability of Mullins−Sekerka type controlled by the gradient field of composition along the growth direction. Application of upward temperature gradient, i.e., higher temperature ahead of the growth front, imposes an additional gradient and brings the following systematic changes in the growth pattern of the dendrites28,29 (Figure 1). First, the upward temperature gradient stabilizes the

2. EXPERIMENTAL SECTION We used isotactic poly(butene-1), iPB-1, kindly supplied by Sun Allomer Ltd. (Mw = 674 000, Mw/Mn = 4.7), poly(vinylidene fluoride), PVDF, of KF1000 (Mw = 2.5 × 105 and Mw/Mn = 2.1, Kureha Chemical Industries, Co., Ltd.), and polyethylene, PE, of NIST SRM 1475 (Mw = 52 000 and Mw/Mn = 2.90). For the crystallization under temperature gradient, polymer samples between a pair of thin glass coverslips (150 μm in thickness, 25 mm in width, and 60 mm in length) were prepared by hot press at 180 °C (iPB-1), 220 °C (PVDF), and 180 °C (PE). The polymer film thickness prepared in this method was about 20 μm. The temperature gradient was applied by two sets of a pair of temperature-controlled bars (9 mm wide). The two sets were placed next to each other and sandwiched the sample glass coverslips between the pair, as shown in Figure 2. The hot bars were heated by a ceramic

Figure 1. Schematic drawing of the effects of temperature gradient on the cellular growth in organic and metallic alloys: larger tip radius, R′ > R, shorter cell groove, h′ < h, and narrower cell width, λ′ < λ.

fluctuations at the tips because the fingering tips experience higher temperature, i.e., lower driving force, and pull back. With this effect, the critical tip radius, R, increases for larger temperature gradient. Second, for the cellular structure, the depth of cell grooves, h, becomes shallower and the cell width, λ, becomes narrower because of the temperature decrease, i.e., higher driving force and faster growth, at the lower side of the grooves. The resultant effects on the cellular pattern are shown as Figure 1. The morphological development of polymer spherulites in the directional solidification under temperature gradient has been studied by Lovinger et al.19,20,23 with the emphasis on the evolution of spherulites from nucleation to directional crystallization in the steady state along the temperature gradient. In the present study, we are concerned with the effects of the externally applied gradient of temperature on the inner structures, such as the spacing of banded crystallization and the patchy pattern in nonbanded crystallization, evolved by lamellar crystallites under temperature gradient in the steady state. In our prior studies, we have experimentally confirmed the proportional relationship between the size of those inner structures, such as the band spacing, P, or the correlation length of the patchy pattern, L, and the size of their building blocks, namely, the width of lamellar crystals at the growth front, λ:

P, L ∝ λ

Figure 2. Schematic drawing of the equipment for the directional crystallization under temperature gradient determined by T1 − T2 and the gap length, g, by moving at a constant speed, V. OM represents an optical microscope. heating plate placed in the bars, and the temperatures of the bars were monitored and controlled with the precision of ±0.1 °C. On the other hand, through the cold bars, temperature-controlled (±0.1 °C) water was circulated. The temperature ranges were 100−240 °C for the hot bars and 5−95 °C for the cold bars. The gap length, g, between the hot and cold bars was in the range 0.3−4.0 mm. The speed of moving corresponds to the crystallization rate of polymers and was in the range 0.5 μm/min to 2 mm/min. For the steady moving at slow speeds required for the slow crystallization of polymers, stick−slip moving was easily activated21 due to the elastic deformation of the system coupled with the friction at the bars and pulleys shown in Figure 2. In order to avoid the stick−slip motion, the steady moving was conducted under large tension by a weight of 3 kg, as shown in Figure 2. For the steady moving under large tension, the system was equipped with a testing machine (Autograph AG-IS, Shimadzu Corp.), with which we could apply the speed down to 0.05 μm/min corresponding to the crystallization rates of majority of polymers. The crystallization was in situ monitored from above the gap of the two hot cells (Figure 2) with a digital optical microscope (KH-7700, Hirox). For the determination of the actual temperature gradient, the relationship between the crystal growth rate of PVDF and temperature was utilized. The position of the growth front, y, along the gradient for

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The relationship indicates that the characteristic size of the inner structures is determined by the width of lamellar crystals, which is controlled by the frequency of instability-driven branching, as shown in eqs 1−3. Therefore, by examining the changes in the inner structures, which can be observed by optical microscopy, we can examine the changes on the lamellar branching at the growth front. If the lamellar width and consequently the inner structures are really determined by the self-induced gradient field, the addition of the external field should have an appreciable influence on the structures. 853


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Figure 3. In situ OM images of PVDF during the directional crystallization under temperature gradient T1 − T2 = 220−80 °C and the gap length g = 1.4 mm. V = (a) 3, (b) 7, (c) 25, and (d) 53 μm/min. The bar represents 400 μm. The arrow indicates the growth direction (the reverse direction of moving), and yi represents the position of the growth front along the temperature gradient. the respective moving speed gave the temperature corresponding to the rate of crystallization at the position, as shown in Figure 3. The relationship between the crystallization temperature and isothermal crystal growth rate was examined with a hot stage (THMS600 controlled by LK-600, Linkam), as shown in Figure 4a. From the

gradient, which is defined by the preset temperature difference, T1 − T2, of the bars and the gap length between them, g, as (dT/dy)set ≡ (T1 − T2)/g. This is due to the finite thickness, h, of the sample + cover glasses, in which temperature gradient along the thickness direction becomes effective and reduces the effective gradient along the moving direction, dT/dy, for the gap length, g, being close to or less than the thickness, h ∼ 320 μm. The typical influence has been simulated, as shown in Figure 5, under the assumption that the hot

Figure 5. Numerical calculation of the temperature in a plate placed between the hot and cold bars kept at 150 and 50 °C, respectively, as in (a). The length/height ratio of the plate is 8. The ratio of the gap, g, and the height, h, is g/h = (b) 0.2 and (c) 4. In (d), the results (dT/ dy)true/(dT/dy)set at the centers are plotted against the ratio g/h.

Figure 4. (a) Linear growth rate, V, of PVDF plotted against crystallization temperature. (b) The shift of the growth fronts, Δy = |yi − y0|, in Figure 3 plotted against crystallization temperature estimated from the V−T relation in (a). (c) The ratio (dT/dy)true/(dT/dy)set plotted against the gap length, g, for various sets of T1 and T2 in the ranges of 100 °C ≤ T1 ≤ 240 °C and 5 °C ≤ T2 ≤ 95 °C.

and cold bars were kept at the set temperatures isothermally, and the thermal contact between the bars and sample was perfect with negligible thermal contact resistance. The actual thermal contact of the sample to the hot and cold bars was maintained by the weight of the upper bars. The application of small amount of silicone oil was effective. The precise control of the contact was difficult especially for large dT/dy. The data scatter in Figure 4c (and in Figures 8, 11, and 14) is probably due to the variation of the contact. After crystallization, the microstructure of the obtained crystals was subsequently examined by polarizing optical microscopy, POM (BX51,

position−temperature relation as in Figure 4b, the inverse of the slope gave the temperature gradient in the cell. It is noted that, as shown in Figure 4c, the true temperature gradient, (dT/dy)true, determined by this method became much smaller than the preset temperature 854


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Figure 6. POM images of iPB-1 crystallized under isothermal conditions (a, d) or under temperature gradient (b, c, e, f) at the respective V and dT/ dy shown at the upper left and right corners. The bars represent 50 μm. The arrow indicates the growth direction. Olympus), and by atomic force microscopy, AFM (SPI3800N, Seiko Instruments Inc.), in a dynamic force mode in air at room temperature. Silicon cantilevers (SI-DF20, Seiko Instruments Inc.) with a resonance frequency of 110−150 kHz were used for the observations. For the AFM observations, samples with free surface were prepared by using spacers (∼50 μm) thicker than the samples and sandwiched between the pair of glass coverslips.

3. RESULTS AND DISCUSSION 3.1. Nonbanded Crystallization of iPB-1 in Form II. Figure 6 shows POM micrographs of iPB-1 crystals obtained isothermally or under temperature gradient. The textures of the nonbanded isothermal crystallization seen under POM in Figure 6a,d are characterized by the existence of a patchy pattern.6 The patterns represent the domains coherent in the orientations and are basically in the shape of inverted triangular shape elongated along the growth direction, i.e., along the radial axis of spherulites under isothermal conditions and the reverse direction of moving under temperature gradient, as schematically shown in Figure 7a. The appearance at the growth front will be clearly due to the competition among growing lamellar crystals. Namely, a growing lamellar crystal can lead to other crystals by fluctuations, which are promoted by the self-induced gradient field and occupy the lateral regions which are originally supposed to be for other lamellar crystals. The winner-take-all competition spontaneously generates the inverted triangular shape of the patchy pattern at the growth front. With the competition, the lamellar crystals in the triangular domains are the descendants of the mother lamella formed by successive branching, and hence the lamellar orientation will keep the correlation among them. This will be the reason why the triangular patchy region becomes evident in the POM images showing the crystal orientations. As is seen in Figure 6, the application of temperature gradient strongly influences the basic pattern with the loss of the triangular domains. The change in the texture can be quantitatively analyzed by introducing the autocorrelation function of the image along the growth direction, i.e., along the long axis of the patchy pattern.6 The results of the correlation length of the autocorrelation function are shown in Figure 8 in the plots against moving speed for the respective dT/dy. Here, the correlation length, L, without the external

Figure 7. Schematic drawing of the growth domains: (a) inverted triangular domains formed by growth front fluctuations and (b) rectangular growth front forced to be aligned on an isothermal (dotted) line normal to the temperature gradient. (c) Typical lateral habit of the rectangular growth of iPB-1 crystal grown from the melt under isothermal condition (AFM amplitude image). 6 The arrow in (a) and (b) indicates the growth direction. The bar in (c) represents 5 μm.

gradient of temperature became longer for slower crystallization speed (i.e., at higher temperatures). The trend is in accordance with the prediction of our modeling of eqs 1−4. Then, the correlation length became shorter with the application of temperature gradient, and the effect became stronger at slower speeds. It means that the patchy pattern became finer, i.e., thinner and shorter, with increasing the gradient, and the influence became more pronounced at slower speed. The loss of triangular patchy pattern with temperature gradient must be a convincing evidence of the growth front fluctuations caused by a self-induced gradient under isothermal conditions and the suppression of the fluctuations by the 855


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3.2. Banded Crystallization of PVDF in the α Form and of PE. 3.2.1. The Observation Results. Figure 10 shows the POM micrographs of PVDF crystals obtained under temperature gradients. As is clearly seen, the band spacing, P, became wider with increasing the gradient. The effect becomes more pronounced at slower speeds, which is more easily recognized in the plot of the band spacing against temperature gradient for the respective speeds in Figure 11a; the data are also replotted in Figure 11b against V for selected dT/dy. Figure 12 shows the AFM images of PVDF crystals on the free surface of the sample. The broader band spacing with higher temperature gradient can also be discernible at the higher magnification. Similar results were obtained for PE crystallization, as shown in Figure 13. With PE, the bands became irregularly deteriorated as well as wider with increasing the gradient. As shown in the plot of the band spacing against speed in Figure 14b, the effects of the moving speed and the gradient were more pronounced with PE: i.e., negligible effects at faster speeds (Figure 13g−i) and broader band spacing at slower speeds with stronger gradient (Figure 13a−c). The data in Figure 14b are also replotted in Figure 14a against dT/dy for selected V. Here, it is noted that, for the irregularly deteriorated patterns (e.g., Figure 13b,c,f) under stronger gradient, the band spacing was determined by the spacing of patchy dark regions along the gradient (shown by double arrows in Figure 13), so that the spacing shown in Figure 14 may be representing the minimum value for the band spacing under high-temperature gradients. 3.2.2. Discussion on the Effects in Comparison with the Results of Nonbanded Crystallization of iPB-1. As the effects of externally applied temperature gradient, the above results of banded crystallization of PVDF and PE showed irregular deterioration of bands with longer spacing. On the other hand, nonbanded crystallization of iPB-1 suggested finer (thinner and shorter) patchy pattern. For the apparent difference in the influences of temperature gradient, we need careful examinations of the effects. First, the irregular deterioration of bands will be related to the disappearance of inverted triangular domains in nonbanded crystallization, the existence of which in banded crystallization is not obvious because of more distinct pattern of bands.34 Since the correlation among descendants of mother lamella in the triangular domains is one of the mechanisms for the coordination of angular phase of twists, it will be reasonable that the loss of the domains results in the deterioration of bands with the application of temperature gradient, which brought shorter coherent region in the phase angle of twist along the tangential direction. From this evidence, we can confirm the strong influences of the external gradient of temperature on the formation of inner structures of both of banded and nonbanded crystallization. Second, in terms of the longer band spacing of PVDF and PE with temperature gradient, we suggest that the correspondence of the proportional relationship of eq 4 between the band spacing and the lamellar width (i.e., wider lamellar width) will be the most probable cause of the change in the inner structure under temperature gradient by the following reasons. First, in our prior experiments on PVDF and PE under isothermal conditions,3−5 we have confirmed the proportional relationship. Second, under temperature gradient, the same trend has been confirmed in the nonbanded crystallization of iPB-1 in terms of the relation between the lamellar width and the characteristic length of the inner structure; i.e., shorter correlation length of patchy pattern, L, with stronger dT/dy was in accordance with

Figure 8. Plots of the correlation length, L, of the patchy pattern in nonbanded crystallization of iPB-1 against the moving speed, V, for iPB-1: isothermal (●) and dT/dy = 20 (▲) and 64 K/mm (■).

application of temperature gradient. Finer patchy pattern of nonbanded crystallization will be the consequence of the temperature gradient, which sets the growth interface being aligned on an isothermal plane normal to the gradient. Since the crystal habit of iPB-1 adheres to the rectangular shape, as shown in Figure 7c, the alignment means a subdivision of the growth front with branching into finer crystals, as shown in Figure 7b. Therefore, the finer patchy pattern corresponds to the cellular structures with narrower and shallower groove of organic and metallic alloys.28,29 In polymer crystallization, the formation of two-dimensional cellular pattern can be realized for the growth from ultrathin films, if the thickness is comparable with the lamellar thickness and the lamellar crystals are basically planar as in the case of iPB-1.33 Figure 9 shows the

Figure 9. Phase-contrast OM images of the cellular growth of iPB-1 crystallized from thin films (68 nm thick) under temperature gradient at the respective V and dT/dy shown at the upper left and right corners. The bars represent 50 μm. The arrow indicates the growth direction. The constant moving was terminated by a quick shift of samples, the boundaries of which are seen at the upper parts.

cellular structure of iPB-1 crystals grown from thin films (68 nm thick) under temperature gradient. The cell width indeed shows the expected change of narrower width under larger temperature gradient for the closely adjoined planar lamellar crystals of iPB-1. 856


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Figure 10. POM images of PVDF crystallized under temperature gradient at the respective V and dT/dy shown at the upper left and right corners, respectively. The bars represent 20 μm. The arrow indicates the growth direction.

Figure 12. AFM amplitude images of PVDF crystallized under temperature gradient at the respective V and dT/dy shown at the upper left and right corners, respectively. The bars represent 5 μm. The arrow indicates the growth direction.

choosing one of the handedness, while in nonbanded crystallization those crystals basically keep their planarity and are supposed to be reoriented in a random direction on the occasion of branching.35 The difference must be due to the nature of surface stresses caused by chain folding in those crystals.36 Therefore, if we may think of the possibility of the violation of the proportional relationship only on banded crystallization under temperature gradient, the violation should suggest the effect of temperature gradient on the surface stresses; i.e., longer band spacing suggests less stresses, and vice versa. For the proportional relationship of eq 4, while the coefficient should be a function of the surface stresses, the constant coefficient has been confirmed experimentally at least for PVDF and PE at several different temperatures. It means that the stresses should be independent (or at least a weak function) of temperature and hence of the gradient. By this reason, the effects of temperature gradient on the stresses will

Figure 11. Plots of the band spacing, P, (a) against the temperature gradient for PVDF at the moving speeds of V = 0.5 (●), 1.0 (▲), 2.0 (■), and 5.0 μm/min (▼) and (b) against the moving speed, V; dT/ dy = 112 (●), 83 (▲), and 50 K/mm (■). The symbol ◆ in (b) is for isothermal crystallization.

narrower cell width, λ, in the growth from ultrathin film, as discussed above. On the other hand, the difference between banded and nonbanded crystallization comes from the manner of twist; lamellar crystals in banded crystallization spontaneously twist 857


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Figure 13. POM images of PE crystallized under temperature gradient at the respective V and dT/dy shown at the upper left and right corners, respectively. The double arrows in (b), (c), and (f) indicate the spacing read from the images. The bars represent 40 μm. The arrow indicates the growth direction.

sustainable structure formation similar to the cellular structure. The unattainable cell growth of band-forming polymers from ultrathin films38 mentioned in the above also supports the argument. Because of this reason, it will not be appropriate to simply apply the existing models of cellular growth of alloys to the banded crystallization. Under temperature gradient, individual lamellar crystal in the banded crystallization probably adjusts the shape for its own sake without the formation of cellular structure. In contrast with the rectangular edge of iPB-1 crystals (Figure 7c), the individual lamellar crystals of PVDF5,39−41 and PE42−45 are with curved edges at the growth front (Figure 15a,b), and the angle at the tips can be variable under different growth conditions.41,44 Then, as shown in Figure 15c, as a consequence of the stabilizing effect of temperature gradient increasing the critical width, wider lamellar width can be reasonably expected with the curved edges. The behavior has a similar corresponding change in the cellular growth of alloys under temperature gradient, where higher temperature gradient leads to larger tip radius of individual dendrites in the solidification of alloys28,29 (Figure 1). Without the constraint of cellular structure, larger tip radius will suggest wider width of individual lamellae. As a possible stabilizing effect of temperature gradient, the following effect is also to be noted. The upward temperature gradient, namely higher temperature in the molten side, means higher diffusion coefficient and/or lower viscosity in the melt. Then, both of faster diffusion and lower viscosity will decrease the gradient and stabilize the interface, e.g., larger diffusion length, D/V, with higher diffusion coefficient. The process will not discriminate banded crystallization from nonbanded, so that the effect will not be considered in the following.

be small enough when we discuss the proportionality between P and λ under temperature gradient. Lastly, the proportional relationship has also been theoretically suggested for both of the cases of continuous twist37 and discontinuous reorientation on the occasion of branching.3 It is noted that the direct confirmation of wider lamellar width for the banded crystallization under temperature gradient was not possible because of the following fundamental reasons. First, the expected width (≤1 μm) is too narrow to be directly observed by optical microscopy, and second, the temperature quench required for subsequent observation of the growth front by AFM or electron microscopy could not be fast enough with the present cells. Third, the examination of the growth from ultrathin films will not be meaningful because the nonplanar lamellar crystals of band-forming polymers inevitably come off from the film with twisting, as classical work suggested,38 in contrast to the cellular growth of basically planar lamellar crystals of iPB-1.33 The proportional relationship suggests wider lamellar width as the cause of the longer band spacing under temperature gradient. For the behavior, we need to recognize the essential difference in the three-dimensional shapes of lamellar crystals as the building blocks of the banded and nonbanded crystallization, i.e., twisting and basically planar lamellae, respectively. The spontaneous twist in banded crystallization means the renewal of geometric configuration surrounding the lamellar crystals all the time in the sense that the three-dimensional coordination of the array of stacked and twisted lamellae is practically impossible. Actually, it will be quite unlikely to expect lamellar platelets placed parallel to each other in a cellular arrangement and twisting in phase in each cell. During the growth from bulk melt, the reconfiguration will prevent the 858


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explained by a simple modeling based on the branching driven by the instability, which is promoted by a self-induced gradient field, as shown in Figure 16a. The temperature gradient, ∂T/∂y,

Figure 14. Plots of the band spacing, P, (a) against the temperature gradient for PE at the moving speeds of V = 12 (●), 64 (▲) and 510 μm/min (■) and (b) against the moving speed, V; dT/dy = 151 (●), 86 (▲), and 53 K/mm (■). The symbol ◆ in (b) is for isothermal crystallization.

Figure 16. (a) Schematic diagram of the gradient in the driving force of the chemical potential for crystallization near the growth front denoted by O without and with the upward temperature gradient for the thin and thick lines, respectively. (b) Plots of P against dT/dy of eq 7 for P0(V) = 100.5 (thin line), 100.75 (dotted line), 101 (broken line), and 101.25 μm (thick line). (c) Plots of P against V of eq 7 for dT/dy = 0 (thin line), 53 (dotted line), 86 (broken line), and 151 K/mm (thick line). It is noted that the coordinate system of (a) is moving with the growth front fixed at the origin, O. In (b) and (c), the data of P0(V) were taken from the experimental results in Figure 14.

corresponds to the gradient in chemical potential, aT, as

aT =

v Δh ∂T ∂Δμ =− s f Tm ∂y ∂y

(5)

because the driving force of the chemical potential, Δμ, is represented by the supercooling, ΔT ≡ Tm − T, as Δμ = vsΔhf(ΔT/Tm), where Δhf represents the heat of fusion per unit volume and Tm the melting point. As seen in eqs 2 and 3, the self-induced gradient field in the chemical potential, a, becomes stronger with increasing crystallization rate, V. On the other hand, the applied external gradient field of temperature weakens the self-induced gradient with the minus sign in eq 5 because of the upward shift of temperature ahead of the growth front, resulting in smaller driving force (Figure 16a). Then, the slope of driving force at

Figure 15. Typical lateral habits with curved edges of (a) PVDF (TEM image)41 and (b) PE (AFM height image)45 single crystals grown from the melt isothermally. (c) Schematic drawing of the expected effects of temperature gradient on nonplanar lamellar crystals with curved edges: larger tip width, λ′ > λ. The bars in (a) and (b) represent 1 μm.

3.2.3. A Possible Modeling of the Effect of Temperature Gradient. As an influence on individual lamellar crystal, the effects of temperature gradient and moving speed can be 859


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In this way, the trend in the dependence of P(λ) on V and ∂T/∂y can be qualitatively explained by this simple modeling. However, the agreement with the experimental results is not good enough. For the quantitative analysis of the growth patterns under gradient fields, we need to model the growth explicitly under the influence of the fields of compositional and/or pressure gradient with moving growth front; the variations of diffusion coefficient, viscosity, and thermal conductivity should also be taken into account. For the purpose, phase field modeling has been often utilized in the studies of cellular and dendritic crystallization in nonpolymeric systems.48,49 For the spherulitic growth of polymers, there have been remarkable developments in the application of phase field modeling recent years.50−52 Future work taking into account of those additional effects is awaited.

the growth front (at the origin, O, in Figure 16a) becomes smaller with the application of temperature gradient. The slope determines the critical lamellar width for branching, as shown in eq 1. The lamellar width is then expected to become wider with the effect, following the expression

⎛ vs γ ⎞1/2 λ = 2π⎜ ⎟ ⎝ a0 + aT ⎠

(6)

in which the slope only with the self-induced gradient, a0, under isothermal condition is replaced by that with the effect of temperature gradient, a0 + aT. It is noted that this approach to the stabilizing effect of temperature gradient is essentially the same as for larger tip radius in the directional solidification of alloys under temperature gradient in the regime of slow moving velocity, as suggested by Kurz and Fisher.28 Here, based on the proportional relationship of eq 4, i.e., P = kλ, the expression of eq 6 predicts the effects on the band spacing, too, as was shown experimentally in the above. Equations 1, 5, and 6 then lead to the following:

1 2

=

1 2

−

Δh f

4. CONCLUSIONS In the present paper, we have experimentally examined the effect of gradient field of temperature on the structural evolution of polymer crystallization from the bulk melt. Under temperature gradient, samples were moved at a constant speed, V, corresponding to the crystallization rate. The temperature gradient then fixes the growth interface at a position, y, of the crystallization temperature, T(y), having V(T(y)). Therefore, the effect basically suppresses the fluctuations of the interface. In our prior studies, we have proposed and experimentally examined the formation mechanism of polymer spherulites by the branching and reorientation of lamellar crystals. In the modeling, the branching is caused by the growth front instability driven by the self-induced gradient field of mass transport in the bulk melt. The temperature gradient is then supposed to suppress the instability and hence should have appreciable influence on the structural evolution of polymer crystallization. In the prior studies under isothermal conditions, we have also confirmed that the width of lamellar crystals, which is limited by the instability-driven branching, determines the band spacing of banded crystallization and the correlation length of patchy pattern of nonbanded crystallization, as the size of the building blocks of those structures. On the basis of the relation, we have experimentally examined the effect of temperature gradient on the characteristic lengths of the inner structures. First, we have confirmed that the temperature gradient suppressed the formation of triangular domains, i.e., the unit of the patchy pattern, and the correlation of the pattern in nonbanded crystallization of iPB-1. The finer structure of patchy pattern corresponds to narrower cell width generally observed for the cellular growth of the array of dendrites of organic and metallic alloys in directional solidification. The cellular growth of iPB-1 from ultrathin films actually confirmed the behavior under temperature gradient. Second, in banded crystallization of PVDF and PE, the bands are irregularly deteriorated with temperature gradient. The deterioration should be related to the loss of triangular domains, in which lamellar orientations of descendants of mother lamellae are correlated with each other. Temperature gradient also enlarged the band spacing. On the basis of the proportional relationship between the band spacing and the lamellar width, it has been suggested that the longer spacing of bands is most probably due to wider lamellar width. The behavior has been ascribed to the stabilizing effect of temperature gradient on the individual lamellae; the effect also has a similar correspondence in the cellular growth of

∂T

2 2

4π k γTm ∂y (7) where P0(V) represents the band spacing for isothermal crystallization with the rate, V. Equation 7 only supposes the basic form of the critical width of eq 1 or 6 and the equivalence of the temperature gradient, ∂T/∂y, and the gradient of chemical potential, aT, represented as eq 5. Therefore, eq 7 is applicable without the knowledge of detailed mechanism of selfinduced gradient field concerning the expression of a0 in eq 6 and simply suggests a stronger effect of ∂T/∂y for longer P0 (smaller P0−2) irrespective of V. With the literature values of PE, i.e., P = 4.5λ,7 γ ≃ 10 erg/cm2, Δhf = 2.8 × 109 erg/cm3, and Tm ≃ 420 K,46 the coefficient in eq 7 is about 8.5 × 10−2 1/(K μm). Figures 16b,c show the plots of eq 7 with the coefficient and P0(V) of isothermal crystallization of PE in Figure 14. Therefore, Figures 16b,c should correspond to Figures 14a,b of PE. The simple modeling qualitatively reproduces the essential features of the experimental results, though the steep rise in P near the critical condition of (a0 + aT) → 0 in eq 6 was not observed experimentally. This simple modeling may exceed the applicability limit near the condition or the deterioration of banding in PE may indicate the divergence of P. In terms of the PVDF results shown in Figure 11, we may also apply the literature values for PVDF, i.e., P = 5.0λ,5 Δhf = 2.0 × 109 erg/cm3, and Tm ≃ 483 K,47 and an empirical relationship46 of γ ≃ 0.1Δhf(ab)1/2 ≃ 10 erg/cm2 with ab = 4.8 × 10−15 cm2 of the cross-sectional area of a stem. The coefficient of eq 7 for PVDF is then estimated to be about 4.2 × 10−2 1/(K μm), not much different from the value of PE. However, as seen in the comparison of the experimental results of Figures 11 and 16b,c with the coefficient of PE, the effect of ∂T/∂y in PVDF is stronger even at much shorter P0. It means that PVDF is more susceptible to temperature gradient. This behavior may be related to the anisotropic lateral shape of PVDF crystals with sharper curved tip along the growth direction (Figure 15a).5,39−41 Narrower tip means smaller tip radius and phenomenologically corresponds to smaller γ in eq 6. If γ in eq 7 becomes one-tenth of the evaluated value, the experimentally observed stronger effect of ∂T/∂y at shorter P0 can be expected from eq 7. P

P0(V )

860


Macromolecules

Article

(31) Bondarenko, Y. A.; Kablov, E. N. Met. Sci. Heat Treat. 2002, 44, 288. (32) Ü stün, E.; Ç adrl, E. C.; Kaya, H. J. Phys.: Condens. Matter 2006, 18, 7825. (33) Kajioka, H.; Taguchi, K.; Toda, A. Macromolecules 2011, 44, 9239. (34) Kajioka, H.; Hoshino, A.; Miyaji, H.; Miyamoto, Y.; Toda, A.; Hikosaka, M. Polymer 2005, 46, 8717. (35) Kajioka, H.; Yoshimoto, S.; Gosh, R. C.; Taguchi, K.; Tanaka, S.; Toda, A. Polymer 2010, 51, 1837. (36) Lotz, B.; Cheng, S. Z. D. Polymer 2005, 46, 577. (37) Hatwalne, Y.; Muthukumar, M. Phys. Rev. Lett. 2010, 105, 107801. (38) Keith, H. D.; Padden, F. J. Jr.; Lotz, B.; Wittmann, J. C. Macromolecules 1989, 22, 2230. (39) Briber, R. M.; Khoury, F. J. Polym. Sci., Part B: Polym. Phys. 1993, 31, 1253. (40) Lovinger, A. J.; Keith, H. D. Macromolecules 1996, 29, 8541. (41) Toda, A.; Arita, T.; Hikosaka, M. Polymer 2001, 42, 2223. (42) Organ, S. J.; Keller, A. J. Mater. Sci. 1985, 20, 1586. (43) Patel, D.; Bassett, D. C. Polymer 2002, 43, 3795. (44) Toda, A. Colloid Polym. Sci. 1992, 270, 667. (45) Toda, A.; Okamura, M.; Hikosaka, M.; Nakagawa, Y. Polymer 2005, 46, 8708. (46) Hoffman, J. D.; Davis, G. T.; Lauritzen Jr., J. I. Treatise on Solid State Chemistry; Plenum Press: New York, 1976; Vol. 3, Chapter 7. (47) Wunderlich, B. Macromolecular Physics; Academic Press: New York, 1980; Vol. 3, p 72. (48) Steinbach, I. Acta Mater. 2008, 56, 4965. (49) Gurevich, S.; Karma, A.; Plapp, M.; Trivedi, R. Phys. Rev. E 2010, 81, 011603. (50) Gránásy, L.; Pusztai, T.; Börzsönyi, T.; Warren, J. A.; Douglas, J. F. Nature Mater. 2004, 3, 645. (51) Gránásy, L.; Pusztai, T.; Tegze, G.; Warren, J. A.; Douglas, J. F. Phys Rev. E 2005, 72, 011605. (52) Xu, H.; Chiu, H.-W.; Okabe, Y.; Kyu, T. Phys Rev. E 2006, 74, 011801.

alloys as a larger tip radius. Those results of nonbanded and banded crystallization hence suggest the strong effects of externally applied gradient field of temperature and the essential role of gradient field in the structural evolution of polymer crystallization.

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AUTHOR INFORMATION

Corresponding Author *Tel +81-82-424-6558; Fax +81-82-424-0757; e-mail atoda@ hiroshima-u.ac.jp.

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ACKNOWLEDGMENTS The authors thank Prof. T. Asano of Shizuoka University for the guidance on the directional solidification under temperature gradient. The authors also thank Prof. S. Tanaka of Hiroshima University and Prof. Y. Yamazaki of Waseda University for helpful discussions. This work was supported by KAKENHI (Grant-in-Aid for Scientific Research) on Priority Area “Soft Matter Physics” from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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REFERENCES

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Structure Evolution in Directional Crystallization of Polymers under

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