Chemical Science
Degree of Supercooling (ΔT ) Dependence and Mass Distribution Function Q(N,t) of Nano-nucleation of Polymers by SAXS
“Nucleation” was assumed to be the early stage of a thin evacuated glass capillary (φ1mm) and then crystallization in classical nucleation theory (CNT) isothermally crystallized at ΔT = 10.5 – 13.0 K. The proposed by Becker and Döring [1] in 1930s. Since SAXS experiment was carried out at beamline it is known that the nucleus has a size on the nm BL40B2. The range of the scattering vector (q) was order, i.e., that nucleus includes 2 – 106 particles or (7– 214) × 10-3 Å-1 and the wavelength (λ) was 1.50 Å. repeating units, we call it a “nano-nucleus.” However, Figure 1 shows typical f(N,t) against t as a it has been too difficult to observe directly nano- parameter of ΔT for N = 2.2 × 104 [rep. unit] which is nucleation because of the technical issues. larger than the size of a critical nano-nucleus (N*= 450 So far, observations with an optical microscope [rep. unit]). The f(N,t) was obtained by applying an (OM) or a bubble chamber have mainly been done to “extended Guinier plot method” to excess scattering trace macroscopic nucleation at a scale larger than intensity IX(q,t) which was obtained by subtracting 1 μm (= macro-crystal). In these studies, it is background intensity [3]. In the figure, it was found assumed that both nano-nucleation and macro- that f(N,t) increases rapidly with an increase of t for crystallization are predominantly controlled by the larger ΔT, while it increases slowly for smaller ΔT. We “critical nano-nucleation,” which is an important but determined τ for each ΔT by the onset time of the unsolved problem. In the CNT, the critical nano- linearly increasing f(N,t), where τ is the induction time nucleation corresponds to the activated state in the for nucleation. free energy of the nucleation process. The inverse of τ is plotted as a function of ΔT -1 We succeeded in direct observation of nano- in Fig. 2. This indicates the relationship as τ-1 ∝ nucleation by small-angle X-ray scattering (SAXS) for exp[–γ/ΔT] (1), where γ is a constant. Thus, it is clear the first time in 2003 [2] and obtained the size that nano-nucleation does not occur when ΔT distribution f(N,t ) of nano-nuclei in 2007 [3], where N becomes significantly small. This indicates that the is the size of a nucleus counted by the number of “induction period” of crystallization is not controlled by repeating units and t is crystallization time. But these a so-called “spinodal decomposition” process [5] but results were obtained only for one case of the degree rather by nucleation process. ΔT dependences of of supercooling, ΔT. In the nucleation study, it is nucleation rate (I) of a macro-crystal and “net flow (j )” important to obtain dependence of nano-nucleation on of nano-nucleation are also shown in Fig. 2. I is ΔT, as ΔT is proportional to the free energy of melting defined by the variation of the number of macro- which is the driving force of nucleation. ΔT is defined crystals per unit volume and time. We have observed as ΔT ≡ Tm°– Tc, where Tm° is the equilibrium melting I of macro-crystals larger than 1 μm by OM and temperature and Tc is crystallization temperature. obtained the experimental formula, I ∝ exp[–C'/ΔT ] In this study, we obtained dependence of nano- (2), where C' is a constant. When the critical nano- nucleation on ΔT and then compared it with the nucleation is the rate-determining process, it is well dependence of macro-crystallization on ΔT to confirm known in CNT that j is given by j ∝ exp[–ΔG*(N*)/kT ] the predominant contribution of the critical nano- ∝ exp[–C/ΔT] (3), where ΔG*(N*) is the free energy of nucleation in macro-crystallization. Second, we t (min) observed the “mass distribution function Q(N,t) ∝ 0 20 40 60 80 Nf(N,t)” directly. The Q(N,t) should exhibit the real 200 × 10-6 1.0 × 10-5 image of nano-nucleation, which cannot be described N = 2.2 × 104 [rep. unit] > N * by f(N,t). Finally, we proposed a new nucleation ΔT = 13.0K 0.8 150 ΔT theory by introducing Q(N,t) in order to solve the = 11.5K ΔT = 10.5K serious problem in CNT that the kinetic equation with 0.6 100 respect to f(N,t) does not satisfy the mass f (N, t) 0.4 f (N, t) conservation law [4]. ΔT : small
We used polyethylene (PE) (NIST, SRM1483a, Mn 50 3 0.2 = 32 × 10 , Mw/Mn =1.1, Tm° = 139.5°C), where Mn and Mw are the number-average and the weight- 0 0.0 average molecular weight, respectively, and Mw/Mn is 0 20 40 60 80 100 the index of dispersion. The nucleating agent of τ t (min) sodium 2,2'-methylene-bis-(4,6-di-t-butylphenylene) phosphate (ADEKA Corp., NA-11SF) was mixed with Fig. 1. Plots of f(N,t) against t as a parameter of ΔT for N, 2.2 × 104 [rep. unit] > N*. It clarified that PE. The sample was melted at 160°C for 5 min within nano-nucleation becomes impossible as ΔT decreases.
108 nucleation process. f(N,t) cannot describe this entire 4 N = 2.2 × 10 [rep. unit] > N * situation at all. Therefore, we focus the discussion on 102 100 τ −1 ∝ e – γ / ΔT phenomena in the range of N ≥ N*. Q(N,t) had a 100 10-2 maximum and became to 0 at Nmax with the increase
) of N for each t, where N is the maximum size of N
) max -1
-2 -4 -1 s 10 10 for a finite time. We obtained Nmax for each t by -3 extrapolating observed f(N,t) to larger N. N m max (min -4 Δ -6 μ
– Cʼ / T -1 10 I ∝ e 10 increases with as t increases. The maximum of Q(N,t) τ μ I ( a ≥ 1 m increased and shifted to larger N with the increase of t. 10-6 10-8 Consequently, we found that the total mass of nano- 10-8 10-10 nuclei increases for N ≥ N*. Our finding indicates that j ∝ e – ΔG*(nano) / kT = e – C / ΔT the crystallinity increases with the increase in t in the 10-10 10-12 nucleation process. We showed that Q(N,t) satisfies 0.06 0.08 0.10 0.12 the mass conservation law which is demanded by the Δ -1 -1 T ( K ) basic equation of a stochastic process [4]. Fig. 2. Plots of τ-1, I and theoretical j against ΔT-1. The observed N of nano-nucleus was N = 2.2 × 104 [rep. unit]>N* and the lateral size of macro-crystal was larger than 1 μm. Since τ-1∝I∝j, it clarified that critical nano- nucleation mainly controls macro-crystallization. 1600 t > 3 × 102 min Qst (N) critical nano-nucleation, ΔG*(N*) ∝ΔT-1 for two- 98 min dimensional nucleus, kT is thermal energy, and C is a 63 min 1200 35 min constant. It is to be noted that j is the theoretical ) 21 min -1 formula. Since τ ∝I∝j (4) was obtained, it clarified 14 min N, t that the critical nano-nucleation mainly controls the ( 800 7 min macro-crystallization. Thus it is concluded that OM is Q a useful tool for convenient routine work in studies of nucleation. 400 Plots of logQ(N,t) (right axis) and logf(N,t) (left axis) against logN as a parameter of t for ΔT = 10.5 K 0 0 are shown in Fig. 3. It was found that f(N,t) decreased 10 102 104 106 108 monotonously but Q(N,t) showed a minimum with a N (rep. unit) magnitude similar to N* and increased with an N* Nmax for t =102 min increase of N for each t. We plotted Q(N,t) against N as a parameter of t in Fig. 4. Plots of Q(N,t) against N as a parameter of t for ΔT Fig. 4. As Q(N,t) showed a minimum with the similar = 10.5 K. The total mass of nano-nuclei for N ≥ N* increases. This behavior indicates that crystallinity magnitude of N* for each t, it was clearly shown that increases with the increase of t in the nucleation process. the critical nano-nucleation is the activated state in the
∝ Qst (N) N Kiyoka Okada a, Sono Sasaki b and Masamichi Hikosaka c,* 4 Qst (N) 8 a Collaborative Research Center, Hiroshima University b SPring-8 / JASRI c Graduate School of Integrated Arts and Sciences,
0 ) 4 Hiroshima University ) Q (N, t) N, t ( f N, t *E-mail: [email protected]
( – 4 log Q fst (N) 0 300 min log 98 min fst (N) References – 8 63 min = const. 35 min [1] V.R. Becker, W. Döring: Ann. Phys. 24 (1935) 719. 21 min – 4 14 min f(N, t) [2] M. Hikosaka et al.: J. Macromol Sci. Phys. B42 (2003) 7 min 847. – 12 0 1 2 3 4 5 6 [3] K. Okada et al.: Polymer 48 (2007) 382. [4] K. Okada, K. Watanabe, A. Toda, S. Sasaki, K. Inoue, * N log N Nmax M. Hikosaka: Polymer 48 (2007) 1116. for t = 14 min [5] M. Imai et al.: Polymer 33 (1992) 4457. Fig. 3. Plots of logQ(N,t) (left axis) and logf(N,t) (right axis) against log N as a parameter of t for ΔT = 10.5 K.
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