108 D De eg gr re ee e o of f S Su up pe er rc co oo ol li in ng g ( ( Δ T T ) ) D De ep pe en nd de en nc ce e a an nd d M Ma as ss s D Di is st tr ri ib bu ut ti io on n F Fu un nc ct ti io on n Q Q( ( N N, ,t t) ) o of f N Na an no o- -n nu uc cl le ea at ti io on n o of f P Po ol ly ym me er rs s b by y S SA AX XS S “Nucleation” was assumed to be the early stage of crystallization in classical nucleation theory (CNT) proposed by Becker and Döring [1] in 1930s. Since it is known that the nucleus has a size on the nm order, i.e., that nucleus includes 2 – 10 6 particles or repeating units, we call it a “nano-nucleus.” However, it has been too difficult to observe directly nano- nucleation because of the technical issues. So far, observations with an optical microscope (OM) or a bubble chamber have mainly been done to trace macroscopic nucleation at a scale larger than 1 μ m (= macro-crystal). In these studies, it is assumed that both nano-nucleation and macro- crystallization are predominantly controlled by the “critical nano-nucleation,” which is an important but unsolved problem. In the CNT, the critical nano- nucleation corresponds to the activated state in the free energy of the nucleation process. We succeeded in direct observation of nano- nucleation by small-angle X-ray scattering (SAXS) for the first time in 2003 [2] and obtained the size distribution f( N,t ) of nano-nuclei in 2007 [3], where N is the size of a nucleus counted by the number of repeating units and t is crystallization time. But these results were obtained only for one case of the degree of supercooling, Δ T. In the nucleation study, it is important to obtain dependence of nano-nucleation on Δ T, as Δ T is proportional to the free energy of melting which is the driving force of nucleation. Δ T is defined as Δ T ≡ T m ° – T c , where T m ° is the equilibrium melting temperature and T c is crystallization temperature. In this study, we obtained dependence of nano- nucleation on Δ T and then compared it with the dependence of macro-crystallization on Δ T to confirm the predominant contribution of the critical nano- nucleation in macro-crystallization. Second, we observed the “mass distribution function Q( N,t) ∝ Nf( N,t)” directly. The Q( N,t) should exhibit the real image of nano-nucleation, which cannot be described by f( N,t). Finally, we proposed a new nucleation theory by introducing Q( N,t) in order to solve the serious problem in CNT that the kinetic equation with respect to f( N,t) does not satisfy the mass conservation law [4]. We used polyethylene (PE) (NIST, SRM1483a, M n = 32 × 10 3 , M w / M n =1.1, T m ° = 139.5°C), where M n and M w are the number-average and the weight- average molecular weight, respectively, and M w / M n is the index of dispersion. The nucleating agent of sodium 2,2'-methylene-bis-(4,6-di-t-butylphenylene) phosphate (ADEKA Corp., NA-11SF) was mixed with PE. The sample was melted at 160°C for 5 min within a thin evacuated glass capillary ( φ 1mm) and then isothermally crystallized at Δ T = 10.5 – 13.0 K. The SAXS experiment was carried out at beamline BL40B2 . The range of the scattering vector ( q) was (7– 214) × 10 -3 Å -1 and the wavelength ( λ ) was 1.50 Å. Figure 1 shows typical f( N,t) against t as a parameter of Δ T for N = 2.2 × 10 4 [rep. unit] which is larger than the size of a critical nano-nucleus ( N*= 450 [rep. unit]). The f( N,t) was obtained by applying an “extended Guinier plot method” to excess scattering intensity I X ( q,t) which was obtained by subtracting background intensity [3]. In the figure, it was found that f( N,t) increases rapidly with an increase of t for larger Δ T, while it increases slowly for smaller Δ T. We determined τ for each Δ T by the onset time of the linearly increasing f( N,t), where τ is the induction time for nucleation. The inverse of τ is plotted as a function of Δ T -1 in Fig. 2. This indicates the relationship as τ -1 ∝ exp[– γ / Δ T] (1), where γ is a constant. Thus, it is clear that nano-nucleation does not occur when Δ T becomes significantly small. This indicates that the “induction period” of crystallization is not controlled by a so-called “spinodal decomposition” process [5] but rather by nucleation process. Δ T dependences of nucleation rate ( I) of a macro-crystal and “net flow ( j )” of nano-nucleation are also shown in Fig. 2. I is defined by the variation of the number of macro- crystals per unit volume and time. We have observed I of macro-crystals larger than 1 μ m by OM and obtained the experimental formula, I ∝ exp[– C'/ Δ T ] (2), where C' is a constant. When the critical nano- nucleation is the rate-determining process, it is well known in CNT that j is given by j ∝ exp[– Δ G*( N*)/ kT ] ∝ exp[– C/ Δ T] (3), where Δ G*( N*) is the free energy of t (min) f (N, t) f (N, t) t (min) τ N = 2.2 × 10 4 [rep. unit] > N * Δ T = 13.0K Δ T = 11.5K Δ T = 10.5K 200 × 10 -6 1.0 × 10 -5 150 100 50 0 80 60 40 20 0 0.8 0.6 0.4 0.2 0.0 100 80 60 40 20 0 Δ T : small Fig. 1. Plots of f ( N,t ) against t as a parameter of Δ T for N , 2.2 × 10 4 [rep. unit] > N * . It clarified that nano-nucleation becomes impossible as Δ T decreases. Chemical Science 109 critical nano-nucleation, Δ G*( N*) ∝Δ T -1 for two- dimensional nucleus, kT is thermal energy, and C is a constant. It is to be noted that j is the theoretical formula. Since τ -1 ∝ I ∝ j (4) was obtained, it clarified that the critical nano-nucleation mainly controls the macro-crystallization. Thus it is concluded that OM is a useful tool for convenient routine work in studies of nucleation. Plots of log Q( N,t) (right axis) and log f( N,t) (left axis) against log N as a parameter of t for Δ T = 10.5 K are shown in Fig. 3. It was found that f( N,t) decreased monotonously but Q( N,t) showed a minimum with a magnitude similar to N* and increased with an increase of N for each t. We plotted Q( N,t) against N as a parameter of t in Fig. 4. As Q( N,t) showed a minimum with the similar magnitude of N* for each t, it was clearly shown that the critical nano-nucleation is the activated state in the nucleation process. f( N,t) cannot describe this entire situation at all. Therefore, we focus the discussion on phenomena in the range of N ≥ N*. Q( N,t) had a maximum and became to 0 at N max with the increase of N for each t, where N max is the maximum size of N for a finite time. We obtained N max for each t by extrapolating observed f( N,t) to larger N. N max increases with as t increases. The maximum of Q( N,t) increased and shifted to larger N with the increase of t. Consequently, we found that the total mass of nano- nuclei increases for N ≥ N*. Our finding indicates that the crystallinity increases with the increase in t in the nucleation process. We showed that Q( N,t) satisfies the mass conservation law which is demanded by the basic equation of a stochastic process [4]. Kiyoka Okada a , Sono Sasaki b and Masamichi Hikosaka c, * a Collaborative Research Center, Hiroshima University b SPring-8 / JASRI c Graduate School of Integrated Arts and Sciences, Hiroshima University *E-mail: hikosaka@hiroshima-u.ac.jp References [1] V.R. Becker, W. Döring: Ann. Phys. 24 (1935) 719. [2] M. Hikosaka et al .: J. Macromol Sci. Phys. B42 (2003) 847. [3] K. Okada et al .: Polymer 48 (2007) 382. [4] K. Okada, K. Watanabe, A. Toda, S. Sasaki, K. Inoue, M. Hikosaka: Polymer 48 (2007) 1116 . [5] M. Imai et al .: Polymer 33 (1992) 4457. 10 2 10 0 10 -2 10 -4 10 -6 10 -8 10 -10 10 0 10 -2 10 -4 10 -6 10 -8 10 -10 10 -12 N = 2.2 × 10 4 [rep. unit] > N * -1 (min-1) τ T -1 ( K -1 ) Δ I ( μ m -3 s -1 ) 0.12 0.10 0.08 0.06 = e I e – C ʼ / Δ T – γ / Δ T – C / Δ T – Δ G* (nano) / kT ∝ τ −1 e ∝ j e ∝ a ≥ 1 μ m Fig. 2. Plots of τ -1 , I and theoretical j against Δ T -1 . The observed N of nano-nucleus was N = 2.2 × 10 4 [rep. unit]> N * and the lateral size of macro-crystal was larger than 1 μ m. Since τ -1 ∝ I ∝ j , it clarified that critical nano- nucleation mainly controls macro-crystallization. N * N max 300 min 98 min 63 min 35 min 21 min 14 min 7 min 8 4 0 – 4 6 5 4 3 2 1 0 – 12 – 8 – 4 0 4 log Q ( N, t ) log f ( N, t ) log N Q ( N, t ) f ( N, t ) Q st ( N ) Q st ( N ) ∝ N f st ( N ) = const. f st ( N ) for t = 14 min Fig. 3. Plots of log Q ( N,t ) (left axis) and log f ( N,t ) (right axis) against log N as a parameter of t for Δ T = 10.5 K. N (rep. unit) 1600 1200 800 400 0 10 0 10 2 10 4 10 6 10 8 7 min 14 min 21 min 35 min 63 min 98 min t > 3 × 10 2 min N * N max f or t =10 2 min Q ( N, t ) Q st ( N ) Fig. 4. Plots of Q ( N,t ) against N as a parameter of t for Δ T = 10.5 K. The total mass of nano-nuclei for N ≥ N * increases. This behavior indicates that crystallinity increases with the increase of t in the nucleation process.
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